Ring (mathematics)

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class="vector-toc-list"> </ul> </li> <li id="toc-Hilbert" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Hilbert"> <div class="vector-toc-text"> 3.2 Hilbert </div> </a> <ul id="toc-Hilbert-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Fraenkel_and_Noether" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Fraenkel_and_Noether"> <div class="vector-toc-text"> 3.3 Fraenkel and Noether </div> </a> <ul id="toc-Fraenkel_and_Noether-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multiplicative_identity_and_the_term_"ring"" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multiplicative_identity_and_the_term_"ring""> <div class="vector-toc-text"> 3.4 Multiplicative identity and the term "ring" </div> </a> <ul id="toc-Multiplicative_identity_and_the_term_"ring"-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Basic_examples" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Basic_examples"> <div class="vector-toc-text"> 4 Basic examples </div> </a> <button aria-controls="toc-Basic_examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Basic examples subsection </button> <ul id="toc-Basic_examples-sublist" class="vector-toc-list"> <li id="toc-Commutative_rings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Commutative_rings"> <div class="vector-toc-text"> 4.1 Commutative rings </div> </a> <ul id="toc-Commutative_rings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Noncommutative_rings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Noncommutative_rings"> <div class="vector-toc-text"> 4.2 Noncommutative rings </div> </a> <ul id="toc-Noncommutative_rings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-rings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-rings"> <div class="vector-toc-text"> 4.3 Non-rings </div> </a> <ul id="toc-Non-rings-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Basic_concepts" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Basic_concepts"> <div class="vector-toc-text"> 5 Basic concepts </div> </a> <button aria-controls="toc-Basic_concepts-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Basic concepts subsection </button> <ul id="toc-Basic_concepts-sublist" class="vector-toc-list"> <li id="toc-Products_and_powers" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Products_and_powers"> <div class="vector-toc-text"> 5.1 Products and powers </div> </a> <ul id="toc-Products_and_powers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Elements_in_a_ring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Elements_in_a_ring"> <div class="vector-toc-text"> 5.2 Elements in a ring </div> </a> <ul id="toc-Elements_in_a_ring-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subring"> <div class="vector-toc-text"> 5.3 Subring </div> </a> <ul id="toc-Subring-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ideal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ideal"> <div class="vector-toc-text"> 5.4 Ideal </div> </a> <ul id="toc-Ideal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Homomorphism" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Homomorphism"> <div class="vector-toc-text"> 5.5 Homomorphism </div> </a> <ul id="toc-Homomorphism-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Quotient_ring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quotient_ring"> <div class="vector-toc-text"> 5.6 Quotient ring </div> </a> <ul id="toc-Quotient_ring-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Module" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Module"> <div class="vector-toc-text"> 6 Module </div> </a> <ul id="toc-Module-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Constructions" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Constructions"> <div class="vector-toc-text"> 7 Constructions </div> </a> <button aria-controls="toc-Constructions-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Constructions subsection </button> <ul id="toc-Constructions-sublist" class="vector-toc-list"> <li id="toc-Direct_product" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Direct_product"> <div class="vector-toc-text"> 7.1 Direct product </div> </a> <ul id="toc-Direct_product-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Polynomial_ring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Polynomial_ring"> <div class="vector-toc-text"> 7.2 Polynomial ring </div> </a> <ul id="toc-Polynomial_ring-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrix_ring_and_endomorphism_ring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matrix_ring_and_endomorphism_ring"> <div class="vector-toc-text"> 7.3 Matrix ring and endomorphism ring </div> </a> <ul id="toc-Matrix_ring_and_endomorphism_ring-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Limits_and_colimits_of_rings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Limits_and_colimits_of_rings"> <div class="vector-toc-text"> 7.4 Limits and colimits of rings </div> </a> <ul id="toc-Limits_and_colimits_of_rings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Localization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Localization"> <div class="vector-toc-text"> 7.5 Localization </div> </a> <ul id="toc-Localization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Completion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Completion"> <div class="vector-toc-text"> 7.6 Completion </div> </a> <ul id="toc-Completion-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rings_with_generators_and_relations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rings_with_generators_and_relations"> <div class="vector-toc-text"> 7.7 Rings with generators and relations </div> </a> <ul id="toc-Rings_with_generators_and_relations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Special_kinds_of_rings" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Special_kinds_of_rings"> <div class="vector-toc-text"> 8 Special kinds of rings </div> </a> <button aria-controls="toc-Special_kinds_of_rings-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Special kinds of rings subsection </button> <ul id="toc-Special_kinds_of_rings-sublist" class="vector-toc-list"> <li id="toc-Domains" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Domains"> <div class="vector-toc-text"> 8.1 Domains </div> </a> <ul id="toc-Domains-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Division_ring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Division_ring"> <div class="vector-toc-text"> 8.2 Division ring </div> </a> <ul id="toc-Division_ring-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Semisimple_rings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semisimple_rings"> <div class="vector-toc-text"> 8.3 Semisimple rings </div> </a> <ul id="toc-Semisimple_rings-sublist" class="vector-toc-list"> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> 8.3.1 Examples </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> 8.3.2 Properties </div> </a> <ul id="toc-Properties-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Central_simple_algebra_and_Brauer_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Central_simple_algebra_and_Brauer_group"> <div class="vector-toc-text"> 8.4 Central simple algebra and Brauer group </div> </a> <ul id="toc-Central_simple_algebra_and_Brauer_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Valuation_ring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Valuation_ring"> <div class="vector-toc-text"> 8.5 Valuation ring </div> </a> <ul id="toc-Valuation_ring-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Rings_with_extra_structure" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Rings_with_extra_structure"> <div class="vector-toc-text"> 9 Rings with extra structure </div> </a> <ul id="toc-Rings_with_extra_structure-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Some_examples_of_the_ubiquity_of_rings" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Some_examples_of_the_ubiquity_of_rings"> <div class="vector-toc-text"> 10 Some examples of the ubiquity of rings </div> </a> <button aria-controls="toc-Some_examples_of_the_ubiquity_of_rings-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Some examples of the ubiquity of rings subsection </button> <ul id="toc-Some_examples_of_the_ubiquity_of_rings-sublist" class="vector-toc-list"> <li id="toc-Cohomology_ring_of_a_topological_space" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cohomology_ring_of_a_topological_space"> <div class="vector-toc-text"> 10.1 Cohomology ring of a topological space </div> </a> <ul id="toc-Cohomology_ring_of_a_topological_space-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Burnside_ring_of_a_group" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Burnside_ring_of_a_group"> <div class="vector-toc-text"> 10.2 Burnside ring of a group </div> </a> <ul id="toc-Burnside_ring_of_a_group-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Representation_ring_of_a_group_ring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Representation_ring_of_a_group_ring"> <div class="vector-toc-text"> 10.3 Representation ring of a group ring </div> </a> <ul id="toc-Representation_ring_of_a_group_ring-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Function_field_of_an_irreducible_algebraic_variety" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Function_field_of_an_irreducible_algebraic_variety"> <div class="vector-toc-text"> 10.4 Function field of an irreducible algebraic variety </div> </a> <ul id="toc-Function_field_of_an_irreducible_algebraic_variety-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Face_ring_of_a_simplicial_complex" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Face_ring_of_a_simplicial_complex"> <div class="vector-toc-text"> 10.5 Face ring of a simplicial complex </div> </a> <ul id="toc-Face_ring_of_a_simplicial_complex-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Category-theoretic_description" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Category-theoretic_description"> <div class="vector-toc-text"> 11 Category-theoretic description </div> </a> <ul id="toc-Category-theoretic_description-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalization" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Generalization"> <div class="vector-toc-text"> 12 Generalization </div> </a> <button aria-controls="toc-Generalization-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Generalization subsection </button> <ul id="toc-Generalization-sublist" class="vector-toc-list"> <li id="toc-Rng" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Rng"> <div class="vector-toc-text"> 12.1 Rng </div> </a> <ul id="toc-Rng-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nonassociative_ring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nonassociative_ring"> <div class="vector-toc-text"> 12.2 Nonassociative ring </div> </a> <ul id="toc-Nonassociative_ring-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Semiring" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Semiring"> <div class="vector-toc-text"> 12.3 Semiring </div> </a> <ul id="toc-Semiring-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Other_ring-like_objects" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Other_ring-like_objects"> <div class="vector-toc-text"> 13 Other ring-like objects </div> </a> <button aria-controls="toc-Other_ring-like_objects-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Other ring-like objects subsection </button> <ul id="toc-Other_ring-like_objects-sublist" class="vector-toc-list"> <li id="toc-Ring_object_in_a_category" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ring_object_in_a_category"> <div class="vector-toc-text"> 13.1 Ring object in a category </div> </a> <ul id="toc-Ring_object_in_a_category-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ring_scheme" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ring_scheme"> <div class="vector-toc-text"> 13.2 Ring scheme </div> </a> <ul id="toc-Ring_scheme-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ring_spectrum" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ring_spectrum"> <div class="vector-toc-text"> 13.3 Ring spectrum </div> </a> <ul id="toc-Ring_spectrum-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 14 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 15 Notes </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 16 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 17 References </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle References subsection </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-General_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#General_references"> <div class="vector-toc-text"> 17.1 General references </div> </a> <ul id="toc-General_references-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Special_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Special_references"> <div class="vector-toc-text"> 17.2 Special references </div> </a> <ul id="toc-Special_references-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Primary_sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Primary_sources"> <div class="vector-toc-text"> 17.3 Primary sources </div> </a> <ul id="toc-Primary_sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Historical_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Historical_references"> <div class="vector-toc-text"> 17.4 Historical references </div> </a> <ul id="toc-Historical_references-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Ring (mathematics)</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 66 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-66" aria-hidden="true" > 66 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AD%D9%84%D9%82%D8%A9_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="حلقة (رياضيات) – Arabic" lang="ar" hreflang="ar" data-title="حلقة (رياضيات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/Kh%C3%B4an" title="Khôan – Minnan" lang="nan" hreflang="nan" data-title="Khôan" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="Minnan" class="interlanguage-link-target">閩南語 / Bân-lâm-gú</a></li><li class="interlanguage-link interwiki-ba mw-list-item"><a href="https://ba.wikipedia.org/wiki/%D2%A0%D1%83%D0%BB%D1%81%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ҡулса (математика) – Bashkir" lang="ba" hreflang="ba" data-title="Ҡулса (математика)" data-language-autonym="Башҡортса" data-language-local-name="Bashkir" class="interlanguage-link-target">Башҡортса</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9A%D0%B0%D0%BB%D1%8C%D1%86%D0%BE_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Кальцо (алгебра) – Belarusian" lang="be" hreflang="be" data-title="Кальцо (алгебра)" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D1%80%D1%8A%D1%81%D1%82%D0%B5%D0%BD_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Пръстен (алгебра) – Bulgarian" lang="bg" hreflang="bg" data-title="Пръстен (алгебра)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Prsten_(matematika)" title="Prsten (matematika) – Bosnian" lang="bs" hreflang="bs" data-title="Prsten (matematika)" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Anell_(matem%C3%A0tiques)" title="Anell (matemàtiques) – Catalan" lang="ca" hreflang="ca" data-title="Anell (matemàtiques)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A3%D0%BD%D0%BA%C4%83_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Ункă (математика) – Chuvash" lang="cv" hreflang="cv" data-title="Ункă (математика)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Okruh_(algebra)" title="Okruh (algebra) – Czech" lang="cs" hreflang="cs" data-title="Okruh (algebra)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Modrwy_(mathemateg)" title="Modrwy (mathemateg) – Welsh" lang="cy" hreflang="cy" data-title="Modrwy (mathemateg)" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Ring_(matematik)" title="Ring (matematik) – Danish" lang="da" hreflang="da" data-title="Ring (matematik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ring_(Algebra)" title="Ring (Algebra) – German" lang="de" hreflang="de" data-title="Ring (Algebra)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ring_(algebra)" title="Ring (algebra) – Estonian" lang="et" hreflang="et" data-title="Ring (algebra)" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B1%CE%BA%CF%84%CF%8D%CE%BB%CE%B9%CE%BF%CF%82_(%CE%AC%CE%BB%CE%B3%CE%B5%CE%B2%CF%81%CE%B1)" title="Δακτύλιος (άλγεβρα) – Greek" lang="el" hreflang="el" data-title="Δακτύλιος (άλγεβρα)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Anillo_(matem%C3%A1tica)" title="Anillo (matemática) – Spanish" lang="es" hreflang="es" data-title="Anillo (matemática)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Ringo_(algebro)" title="Ringo (algebro) – Esperanto" lang="eo" hreflang="eo" data-title="Ringo (algebro)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Eraztun_(matematika)" title="Eraztun (matematika) – Basque" lang="eu" hreflang="eu" data-title="Eraztun (matematika)" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D9%84%D9%82%D9%87_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="حلقه (ریاضیات) – Persian" lang="fa" hreflang="fa" data-title="حلقه (ریاضیات)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Anneau_(math%C3%A9matiques)" title="Anneau (mathématiques) – French" lang="fr" hreflang="fr" data-title="Anneau (mathématiques)" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/F%C3%A1inne_(matamaitic)" title="Fáinne (matamaitic) – Irish" lang="ga" hreflang="ga" data-title="Fáinne (matamaitic)" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Anel_(%C3%A1lxebra)" title="Anel (álxebra) – Galician" lang="gl" hreflang="gl" data-title="Anel (álxebra)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학) – Korean" lang="ko" hreflang="ko" data-title="환 (수학)" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%95%D5%B2%D5%A1%D5%AF_(%D5%B4%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1)" title="Օղակ (մաթեմատիկա) – Armenian" lang="hy" hreflang="hy" data-title="Օղակ (մաթեմատիկա)" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Prsten_(matematika)" title="Prsten (matematika) – Croatian" lang="hr" hreflang="hr" data-title="Prsten (matematika)" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Gelanggang_(matematika)" title="Gelanggang (matematika) – Indonesian" lang="id" hreflang="id" data-title="Gelanggang (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Anello_(algebra)" title="Anello (algebra) – Interlingua" lang="ia" hreflang="ia" data-title="Anello (algebra)" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Anello_(algebra)" title="Anello (algebra) – Italian" lang="it" hreflang="it" data-title="Anello (algebra)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%97%D7%95%D7%92_(%D7%9E%D7%91%D7%A0%D7%94_%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%99)" title="חוג (מבנה אלגברי) – Hebrew" lang="he" hreflang="he" data-title="חוג (מבנה אלגברי)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B0%E0%B2%BF%E0%B2%82%E0%B2%97%E0%B3%8D" title="ರಿಂಗ್ – Kannada" lang="kn" hreflang="kn" data-title="ರಿಂಗ್" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%A0%E1%83%92%E1%83%9D%E1%83%9A%E1%83%98_(%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90)" title="რგოლი (მათემატიკა) – Georgian" lang="ka" hreflang="ka" data-title="რგოლი (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target">ქართული</a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Anellus" title="Anellus – Latin" lang="la" hreflang="la" data-title="Anellus" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target">Latina</a></li><li class="interlanguage-link interwiki-lb mw-list-item"><a href="https://lb.wikipedia.org/wiki/Rank_(Algeber)" title="Rank (Algeber) – Luxembourgish" lang="lb" hreflang="lb" data-title="Rank (Algeber)" data-language-autonym="Lëtzebuergesch" data-language-local-name="Luxembourgish" class="interlanguage-link-target">Lëtzebuergesch</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Anell_(matematega)" title="Anell (matematega) – Lombard" lang="lmo" hreflang="lmo" data-title="Anell (matematega)" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Gy%C5%B1r%C5%B1_(matematika)" title="Gyűrű (matematika) – Hungarian" lang="hu" hreflang="hu" data-title="Gyűrű (matematika)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%B5%E0%B4%B2%E0%B4%AF%E0%B4%82_(%E0%B4%97%E0%B4%A3%E0%B4%BF%E0%B4%A4%E0%B4%82)" title="വലയം (ഗണിതം) – Malayalam" lang="ml" hreflang="ml" data-title="വലയം (ഗണിതം)" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B0%E0%A4%BF%E0%A4%82%E0%A4%97" title="रिंग – Marathi" lang="mr" hreflang="mr" data-title="रिंग" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Gelanggang_(matematik)" title="Gelanggang (matematik) – Malay" lang="ms" hreflang="ms" data-title="Gelanggang (matematik)" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Ring_(wiskunde)" title="Ring (wiskunde) – Dutch" lang="nl" hreflang="nl" data-title="Ring (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%92%B0_(%E6%95%B0%E5%AD%A6)" title="環 (数学) – Japanese" lang="ja" hreflang="ja" data-title="環 (数学)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Ring_(matematikk)" title="Ring (matematikk) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Ring (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Ring_i_matematikk" title="Ring i matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Ring i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-nov mw-list-item"><a href="https://nov.wikipedia.org/wiki/Ringe_(matematike)" title="Ringe (matematike) – Novial" lang="nov" hreflang="nov" data-title="Ringe (matematike)" data-language-autonym="Novial" data-language-local-name="Novial" class="interlanguage-link-target">Novial</a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Anel" title="Anel – Piedmontese" lang="pms" hreflang="pms" data-title="Anel" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target">Piemontèis</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Pier%C5%9Bcie%C5%84_(matematyka)" title="Pierścień (matematyka) – Polish" lang="pl" hreflang="pl" data-title="Pierścień (matematyka)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Anel_(matem%C3%A1tica)" title="Anel (matemática) – Portuguese" lang="pt" hreflang="pt" data-title="Anel (matemática)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Inel_(matematic%C4%83)" title="Inel (matematică) – Romanian" lang="ro" hreflang="ro" data-title="Inel (matematică)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BB%D1%8C%D1%86%D0%BE_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Кольцо (математика) – Russian" lang="ru" hreflang="ru" data-title="Кольцо (математика)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Aneddu_(matim%C3%A0tica)" title="Aneddu (matimàtica) – Sicilian" lang="scn" hreflang="scn" data-title="Aneddu (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="Sicilian" class="interlanguage-link-target">Sicilianu</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Ring_(mathematics)" title="Ring (mathematics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Ring (mathematics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Okruh_(algebra)" title="Okruh (algebra) – Slovak" lang="sk" hreflang="sk" data-title="Okruh (algebra)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kolobar_(algebra)" title="Kolobar (algebra) – Slovenian" lang="sl" hreflang="sl" data-title="Kolobar (algebra)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D0%B0%D1%80%D1%81%D0%BA%D0%B8_%D0%BF%D1%80%D1%81%D1%82%D0%B5%D0%BD" title="Алгебарски прстен – Serbian" lang="sr" hreflang="sr" data-title="Алгебарски прстен" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Algebarski_prsten" title="Algebarski prsten – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Algebarski prsten" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Rengas_(matematiikka)" title="Rengas (matematiikka) – Finnish" lang="fi" hreflang="fi" data-title="Rengas (matematiikka)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ring_(matematik)" title="Ring (matematik) – Swedish" lang="sv" hreflang="sv" data-title="Ring (matematik)" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%B3%E0%AF%88%E0%AE%AF%E0%AE%AE%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="வளையம் (கணிதம்) – Tamil" lang="ta" hreflang="ta" data-title="வளையம் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%A3%E0%B8%B4%E0%B8%87_(%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="ริง (คณิตศาสตร์) – Thai" lang="th" hreflang="th" data-title="ริง (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Halka" title="Halka – Turkish" lang="tr" hreflang="tr" data-title="Halka" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9A%D1%96%D0%BB%D1%8C%D1%86%D0%B5_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Кільце (алгебра) – Ukrainian" lang="uk" hreflang="uk" data-title="Кільце (алгебра)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AD%D9%84%D9%82%DB%81_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="حلقہ (ریاضی) – Urdu" lang="ur" hreflang="ur" data-title="حلقہ (ریاضی)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/V%C3%A0nh" title="Vành – Vietnamese" lang="vi" hreflang="vi" data-title="Vành" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Algebraic structure with addition and multiplication</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is 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.sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist" style="width: 20.5em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → Ring theory <a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Basic concepts</div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a class="mw-selflink selflink">Rings</a> <dl><dd>• <a href="/wiki/Subring" title="Subring">Subrings</a></dd> <dd>• <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">Ideal</a></dd> <dd>• <a href="/wiki/Quotient_ring" title="Quotient ring">Quotient ring</a> <dl><dd>• <a href="/wiki/Fractional_ideal" title="Fractional ideal">Fractional ideal</a></dd> <dd>• <a href="/wiki/Total_ring_of_fractions" title="Total ring of fractions">Total ring of fractions</a></dd></dl></dd> <dd>• <a href="/wiki/Product_of_rings" title="Product of rings">Product of rings</a></dd> <dd>• <a href="/wiki/Free_product_of_associative_algebras" title="Free product of associative algebras">Free product of associative algebras</a></dd> <dd>• <a href="/wiki/Tensor_product_of_algebras" title="Tensor product of algebras">Tensor product of algebras</a></dd></dl> <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">Ring homomorphisms</a> <dl><dd>• <a href="/wiki/Kernel_(algebra)#Ring_homomorphisms" title="Kernel (algebra)">Kernel</a></dd> <dd>• <a href="/wiki/Inner_automorphism#Ring_case" title="Inner automorphism">Inner automorphism</a></dd> <dd>• <a href="/wiki/Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius endomorphism</a></dd></dl> <a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structures</a> <dl><dd>• <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a></dd> <dd>• <a href="/wiki/Associative_algebra" title="Associative algebra">Associative algebra</a></dd> <dd>• <a href="/wiki/Graded_ring" title="Graded ring">Graded ring</a></dd> <dd>• <a href="/wiki/Involutive_ring" class="mw-redirect" title="Involutive ring">Involutive ring</a></dd> <dd>• <a href="/wiki/Category_of_rings" title="Category of rings">Category of rings</a> <dl><dd>• <a href="/wiki/Integer" title="Integer">Initial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></dd> <dd>• <a href="/wiki/Zero_ring" title="Zero ring">Terminal ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45ab495cb8cfbac68a9322af662c3d6c7dbe494" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.686ex; height:2.843ex;" alt="{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }"></dd></dl></dd></dl> Related structures <dl><dd>• <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> <dl><dd>• <a href="/wiki/Finite_field" title="Finite field">Finite field</a></dd></dl></dd> <dd>• <a href="/wiki/Non-associative_ring" class="mw-redirect" title="Non-associative ring">Non-associative ring</a> <dl><dd>• <a href="/wiki/Lie_ring" class="mw-redirect" title="Lie ring">Lie ring</a></dd> <dd>• <a href="/wiki/Jordan_ring" class="mw-redirect" title="Jordan ring">Jordan ring</a></dd></dl></dd> <dd>• <a href="/wiki/Semiring" title="Semiring">Semiring</a> <dl><dd>• <a href="/wiki/Semifield" title="Semifield">Semifield</a></dd></dl></dd></dl></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="/wiki/Commutative_ring" title="Commutative ring">Commutative rings</a> <dl><dd>• <a href="/wiki/Integral_domain" title="Integral domain">Integral domain</a> <dl><dd>• <a href="/wiki/Integrally_closed_domain" title="Integrally closed domain">Integrally closed domain</a></dd> <dd>• <a href="/wiki/GCD_domain" title="GCD domain">GCD domain</a></dd> <dd>• <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">Unique factorization domain</a></dd> <dd>• <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">Principal ideal domain</a></dd> <dd>• <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domain</a></dd> <dd>• <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> <dl><dd>• <a href="/wiki/Finite_field" title="Finite field">Finite field</a></dd></dl></dd> <dd>• <a href="/wiki/Polynomial_ring" title="Polynomial ring">Polynomial ring</a></dd> <dd>• <a href="/wiki/Formal_power_series_ring" class="mw-redirect" title="Formal power series ring">Formal power series ring</a></dd></dl></dd></dl> <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a> <dl><dd>• <a href="/wiki/Algebraic_number_field" title="Algebraic number field">Algebraic number field</a></dd> <dd>• <a href="/wiki/Integers_modulo_n" class="mw-redirect" title="Integers modulo n">Integers modulo n</a></dd> <dd>• <a href="/wiki/Ring_of_integers" title="Ring of integers">Ring of integers</a></dd> <dd>• <a href="/wiki/P-adic_integer" class="mw-redirect" title="P-adic integer">p-adic integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}"></dd> <dd>• <a href="/wiki/P-adic_number" title="P-adic number">p-adic numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"></dd> <dd>• <a href="/wiki/Pr%C3%BCfer_group#The_Prüfer_group_as_a_ring" title="Prüfer group">Prüfer p-ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} (p^{\infty })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} (p^{\infty })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14af623e08c241266c125ad927dd35086ec8ce90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.404ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} (p^{\infty })}"></dd></dl></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Noncommutative_algebra" class="mw-redirect" title="Noncommutative algebra">Noncommutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="/wiki/Noncommutative_ring" title="Noncommutative ring">Noncommutative rings</a> <dl><dd>• <a href="/wiki/Division_ring" title="Division ring">Division ring</a></dd> <dd>• <a href="/wiki/Semiprimitive_ring" title="Semiprimitive ring">Semiprimitive ring</a></dd> <dd>• <a href="/wiki/Simple_ring" title="Simple ring">Simple ring</a></dd> <dd>• <a href="/wiki/Commutator_(ring_theory)" class="mw-redirect" title="Commutator (ring theory)">Commutator</a></dd></dl> <a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a> <a href="/wiki/Free_algebra" title="Free algebra">Free algebra</a> <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> <dl><dd>• <a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></dd></dl> <a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Ring_theory_sidebar" title="Template:Ring theory sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Ring_theory_sidebar" title="Template talk:Ring theory sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Ring_theory_sidebar" title="Special:EditPage/Template:Ring theory sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, rings are <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structures</a> that generalize <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>: multiplication need not be <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a> and <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverses</a> need not exist. Informally, a ring is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> equipped with two <a href="/wiki/Binary_operation" title="Binary operation">binary operations</a> satisfying properties analogous to those of <a href="/wiki/Addition" title="Addition">addition</a> and <a href="/wiki/Multiplication" title="Multiplication">multiplication</a> of <a href="/wiki/Integer" title="Integer">integers</a>. Ring <a href="/wiki/Element_(mathematics)" title="Element (mathematics)">elements</a> may be numbers such as integers or <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, but they may also be non-numerical objects such as <a href="/wiki/Polynomial" title="Polynomial">polynomials</a>, <a href="/wiki/Square_matrices" class="mw-redirect" title="Square matrices">square matrices</a>, <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>, and <a href="/wiki/Power_series" title="Power series">power series</a>. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1246091330"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="display:block;margin-bottom:0.35em;"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structures</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Group</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Group</a></li> <li><a href="/wiki/Semigroup" title="Semigroup">Semigroup</a> / <a href="/wiki/Monoid" title="Monoid">Monoid</a></li> <li><a href="/wiki/Racks_and_quandles" title="Racks and quandles">Rack and quandle</a></li> <li><a href="/wiki/Quasigroup" title="Quasigroup">Quasigroup and loop</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Abelian_group" title="Abelian group">Abelian group</a></li> <li><a href="/wiki/Magma_(algebra)" title="Magma (algebra)">Magma</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a></li></ul> </div> <a href="/wiki/Group_theory" title="Group theory">Group theory</a></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a class="mw-selflink selflink">Ring</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a class="mw-selflink selflink">Ring</a></li> <li><a href="/wiki/Rng_(algebra)" title="Rng (algebra)">Rng</a></li> <li><a href="/wiki/Semiring" title="Semiring">Semiring</a></li> <li><a href="/wiki/Near-ring" title="Near-ring">Near-ring</a></li> <li><a href="/wiki/Commutative_ring" title="Commutative ring">Commutative ring</a></li> <li><a href="/wiki/Domain_(ring_theory)" title="Domain (ring theory)">Domain</a></li> <li><a href="/wiki/Integral_domain" title="Integral domain">Integral domain</a></li> <li><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a></li> <li><a href="/wiki/Division_ring" title="Division ring">Division ring</a></li> <li><a href="/wiki/Lie_algebra#Lie_ring" title="Lie algebra">Lie ring</a></li></ul> </div> <a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a></li> <li><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></li> <li><a href="/wiki/Complemented_lattice" title="Complemented lattice">Complemented lattice</a></li> <li><a href="/wiki/Total_order" title="Total order">Total order</a></li> <li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a></li></ul> </div> <ul><li><a href="/wiki/Map_of_lattices" title="Map of lattices">Map of lattices</a></li> <li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice theory</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a></li> <li><a href="/wiki/Group_with_operators" title="Group with operators">Group with operators</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li></ul> </div> <ul><li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Algebra_over_a_field" title="Algebra over a field">Algebra</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Algebra_over_a_field" title="Algebra over a field">Algebra</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Associative_algebra" title="Associative algebra">Associative</a></li> <li><a href="/wiki/Non-associative_algebra" title="Non-associative algebra">Non-associative</a></li> <li><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebra</a></li> <li><a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a></li> <li><a href="/wiki/Graded_ring" title="Graded ring">Graded</a></li> <li><a href="/wiki/Bialgebra" title="Bialgebra">Bialgebra</a></li> <li><a href="/wiki/Hopf_algebra" title="Hopf algebra">Hopf algebra</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Algebraic_structures" title="Template:Algebraic structures"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Algebraic_structures" title="Template talk:Algebraic structures"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Algebraic_structures" title="Special:EditPage/Template:Algebraic structures"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> Formally, a ring is a set endowed with two binary operations called addition and multiplication such that the ring is an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> with respect to the addition operator, and the multiplication operator is <a href="/wiki/Associative_property" title="Associative property">associative</a>, is <a href="/wiki/Distributive_property" title="Distributive property">distributive</a> over the addition operation, and has a multiplicative <a href="/wiki/Identity_element" title="Identity element">identity element</a>. (Some authors define rings without requiring a multiplicative identity and instead call the structure defined above a ring with identity. See <a href="#Variations_on_the_definition">§ Variations on the definition</a>.) Whether a ring is commutative has profound implications on its behavior. <a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative algebra</a>, the theory of <a href="/wiki/Commutative_ring" title="Commutative ring">commutative rings</a>, is a major branch of <a href="/wiki/Ring_theory" title="Ring theory">ring theory</a>. Its development has been greatly influenced by problems and ideas of <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a> and <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. The simplest commutative rings are those that admit division by non-zero elements; such rings are called <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>. Examples of commutative rings include the set of integers with their standard addition and multiplication, the set of polynomials with their addition and multiplication, the <a href="/wiki/Coordinate_ring" class="mw-redirect" title="Coordinate ring">coordinate ring</a> of an <a href="/wiki/Affine_algebraic_variety" class="mw-redirect" title="Affine algebraic variety">affine algebraic variety</a>, and the <a href="/wiki/Ring_of_integers" title="Ring of integers">ring of integers</a> of a number field. Examples of noncommutative rings include the ring of n × n real <a href="/wiki/Square_matrix" title="Square matrix">square matrices</a> with n ≥ 2, <a href="/wiki/Group_ring" title="Group ring">group rings</a> in <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a>, <a href="/wiki/Operator_algebra" title="Operator algebra">operator algebras</a> in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>, <a href="/wiki/Ring_of_differential_operators" class="mw-redirect" title="Ring of differential operators">rings of differential operators</a>, and <a href="/wiki/Cohomology_ring" title="Cohomology ring">cohomology rings</a> in <a href="/wiki/Topology" title="Topology">topology</a>. The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Dedekind</a>, <a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert</a>, <a href="/wiki/Abraham_Fraenkel" title="Abraham Fraenkel">Fraenkel</a>, and <a href="/wiki/Emmy_Noether" title="Emmy Noether">Noether</a>. Rings were first formalized as a generalization of <a href="/wiki/Dedekind_domain" title="Dedekind domain">Dedekind domains</a> that occur in <a href="/wiki/Number_theory" title="Number theory">number theory</a>, and of <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial rings</a> and rings of invariants that occur in <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> and <a href="/wiki/Invariant_theory" title="Invariant theory">invariant theory</a>. They later proved useful in other branches of mathematics such as <a href="/wiki/Geometry" title="Geometry">geometry</a> and <a href="/wiki/Mathematical_analysis" title="Mathematical analysis">analysis</a>. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=1" title="Edit section: Definition">edit</a>]</div> A ring is a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> R equipped with two binary operations<a href="#cite_note-1">[a]</a> + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms:<a href="#cite_note-FOOTNOTEBourbaki198996Ch_1,_§8.1-2">[1]</a><a href="#cite_note-FOOTNOTEMac_LaneBirkhoff196785-3">[2]</a><a href="#cite_note-FOOTNOTELang200283-4">[3]</a> <ol><li>R is an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> under addition, meaning that: <ul><li>(a + b) + c = a + (b + c) for all a, b, c in R (that is, + is <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associative</a>).</li> <li>a + b = b + a for all a, b in R (that is, + is <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutative</a>).</li> <li>There is an element 0 in R such that a + 0 = a for all a in R (that is, 0 is the <a href="/wiki/Additive_identity" title="Additive identity">additive identity</a>).</li> <li>For each a in R there exists −a in R such that a + (−a) = 0 (that is, −a is the <a href="/wiki/Additive_inverse" title="Additive inverse">additive inverse</a> of a).</li></ul></li> <li>R is a <a href="/wiki/Monoid" title="Monoid">monoid</a> under multiplication, meaning that: <ul><li>(a · b) · c = a · (b · c) for all a, b, c in R (that is, ⋅ is associative).</li> <li>There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R (that is, 1 is the <a href="/wiki/Multiplicative_identity" class="mw-redirect" title="Multiplicative identity">multiplicative identity</a>).<a href="#cite_note-5">[b]</a></li></ul></li> <li>Multiplication is <a href="/wiki/Distributive_law" class="mw-redirect" title="Distributive law">distributive</a> with respect to addition, meaning that: <ul><li>a · (b + c) = (a · b) + (a · c) for all a, b, c in R (left distributivity).</li> <li>(b + c) · a = (b · a) + (c · a) for all a, b, c in R (right distributivity).</li></ul></li></ol> In notation, the multiplication symbol · is often omitted, in which case a · b is written as ab. <div class="mw-heading mw-heading3"><h3 id="Variations_on_the_definition">Variations on the definition</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=2" title="Edit section: Variations on the definition">edit</a>]</div> In the terminology of this article, a ring is defined to have a multiplicative identity, while a structure with the same axiomatic definition but without the requirement for a multiplicative identity is instead called a "<a href="/wiki/Rng_(algebra)" title="Rng (algebra)">rng</a>" (IPA: <a href="/wiki/Help:IPA/English" title="Help:IPA/English">/rʊŋ/</a>) with a missing "i". For example, the set of <a href="/wiki/Even_integer" class="mw-redirect" title="Even integer">even integers</a> with the usual + and ⋅ is a rng, but not a ring. As explained in <a href="#History">§ History</a> below, many authors apply the term "ring" without requiring a multiplicative identity. Although ring addition is <a href="/wiki/Commutative_law" class="mw-redirect" title="Commutative law">commutative</a>, ring multiplication is not required to be commutative: ab need not necessarily equal ba. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called <a href="/wiki/Commutative_ring" title="Commutative ring">commutative rings</a>. Books on commutative algebra or algebraic geometry often adopt the convention that ring means commutative ring, to simplify terminology. In a ring, multiplicative inverses are not required to exist. A non<a href="/wiki/Zero_ring" title="Zero ring">zero</a> commutative ring in which every nonzero element has a <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a> is called a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>. The additive group of a ring is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms.<a href="#cite_note-FOOTNOTEIsaacs1994160-6">[4]</a> The proof makes use of the "1", and does not work in a rng. (For a rng, omitting the axiom of commutativity of addition leaves it inferable from the remaining rng assumptions only for elements that are products: ab + cd = cd + ab.) There are a few authors who use the term "ring" to refer to structures in which there is no requirement for multiplication to be associative.<a href="#cite_note-7">[5]</a> For these authors, every <a href="/wiki/Algebra_over_a_field" title="Algebra over a field">algebra</a> is a "ring". <div class="mw-heading mw-heading2"><h2 id="Illustration">Illustration</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=3" title="Edit section: Illustration">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Number-line.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Number-line.svg/410px-Number-line.svg.png" decoding="async" width="410" height="27" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/93/Number-line.svg/615px-Number-line.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/93/Number-line.svg/820px-Number-line.svg.png 2x" data-file-width="750" data-file-height="50" /></a><figcaption>The <a href="/wiki/Integer" title="Integer">integers</a>, along with the two operations of <a href="/wiki/Addition" title="Addition">addition</a> and <a href="/wiki/Multiplication" title="Multiplication">multiplication</a>, form the prototypical example of a ring.</figcaption></figure> The most familiar example of a ring is the set of all integers ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3aa4cb112cbe4f94a3ff8569f869c31dce5fce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.197ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ,}">⁠ consisting of the <a href="/wiki/Number" title="Number">numbers</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \dots ,-5,-4,-3,-2,-1,0,1,2,3,4,5,\dots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>…</mo> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \dots ,-5,-4,-3,-2,-1,0,1,2,3,4,5,\dots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cfd12a48efb912d354df2a4b13958229538efd27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:40.069ex; height:2.509ex;" alt="{\displaystyle \dots ,-5,-4,-3,-2,-1,0,1,2,3,4,5,\dots }"></dd></dl> The axioms of a ring were elaborated as a generalization of familiar properties of addition and multiplication of integers. <div class="mw-heading mw-heading3"><h3 id="Some_properties">Some properties</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=4" title="Edit section: Some properties">edit</a>]</div> Some basic properties of a ring follow immediately from the axioms: <ul><li>The additive identity is unique.</li> <li>The additive inverse of each element is unique.</li> <li>The multiplicative identity is unique.</li> <li>For any element x in a ring R, one has x0 = 0 = 0x (zero is an <a href="/wiki/Absorbing_element" title="Absorbing element">absorbing element</a> with respect to multiplication) and (–1)x = –x.</li> <li>If 0 = 1 in a ring R (or more generally, 0 is a unit element), then R has only one element, and is called the <a href="/wiki/Zero_ring" title="Zero ring">zero ring</a>.</li> <li>If a ring R contains the zero ring as a subring, then R itself is the zero ring.<a href="#cite_note-FOOTNOTEIsaacs1994161-8">[6]</a></li> <li>The <a href="/wiki/Binomial_formula" class="mw-redirect" title="Binomial formula">binomial formula</a> holds for any x and y satisfying xy = yx.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Example:_Integers_modulo_4">Example: Integers modulo 4</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=5" title="Edit section: Example: Integers modulo 4">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">Modular arithmetic</a></div> Equip the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /4\mathbb {Z} =\left\{{\overline {0}},{\overline {1}},{\overline {2}},{\overline {3}}\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>0</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>1</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>2</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo accent="false">¯</mo> </mover> </mrow> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /4\mathbb {Z} =\left\{{\overline {0}},{\overline {1}},{\overline {2}},{\overline {3}}\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d75fb08c8e035122a0d6bed3e3513fcb24447cdb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.836ex; height:4.843ex;" alt="{\displaystyle \mathbb {Z} /4\mathbb {Z} =\left\{{\overline {0}},{\overline {1}},{\overline {2}},{\overline {3}}\right\}}"> with the following operations: <ul><li>The sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {x}}+{\overline {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {x}}+{\overline {y}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e6d71e60fdb7117c681854580a4bdebb0fb730f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.566ex; height:2.676ex;" alt="{\displaystyle {\overline {x}}+{\overline {y}}}"> in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /4\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /4\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3ded1f832ca4738a18c9e4779381bd9591c058" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /4\mathbb {Z} }">⁠ is the remainder when the integer x + y is divided by 4 (as x + y is always smaller than 8, this remainder is either x + y or x + y − 4). For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {2}}+{\overline {3}}={\overline {1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>2</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>1</mn> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {2}}+{\overline {3}}={\overline {1}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/797a15b50e32954f45330f42720bc3d016214b8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.771ex; height:3.009ex;" alt="{\displaystyle {\overline {2}}+{\overline {3}}={\overline {1}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {3}}+{\overline {3}}={\overline {2}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>2</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {3}}+{\overline {3}}={\overline {2}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f542bb21871274e7f0cb2d353c1257704a65623" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.418ex; height:3.009ex;" alt="{\displaystyle {\overline {3}}+{\overline {3}}={\overline {2}}.}"></li> <li>The product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {x}}\cdot {\overline {y}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {x}}\cdot {\overline {y}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50794127a87517f127ad072dd45bee54fffcaed6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.405ex; height:2.676ex;" alt="{\displaystyle {\overline {x}}\cdot {\overline {y}}}"> in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /4\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /4\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3ded1f832ca4738a18c9e4779381bd9591c058" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /4\mathbb {Z} }">⁠ is the remainder when the integer xy is divided by 4. For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {2}}\cdot {\overline {3}}={\overline {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>2</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>2</mn> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {2}}\cdot {\overline {3}}={\overline {2}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ab24b33a90fe702102a430ea7fe1c6924666fab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.61ex; height:2.843ex;" alt="{\displaystyle {\overline {2}}\cdot {\overline {3}}={\overline {2}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {3}}\cdot {\overline {3}}={\overline {1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>1</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {3}}\cdot {\overline {3}}={\overline {1}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12fe1b546fb9d2cfed9b69e2dc0dd01fd5896565" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.257ex; height:2.843ex;" alt="{\displaystyle {\overline {3}}\cdot {\overline {3}}={\overline {1}}.}"></li></ul> Then ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /4\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /4\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3ded1f832ca4738a18c9e4779381bd9591c058" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /4\mathbb {Z} }">⁠ is a ring: each axiom follows from the corresponding axiom for ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89f4f38f32c2068bca9dc701d13b03dd4a5d52ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} .}">⁠ If x is an integer, the remainder of x when divided by 4 may be considered as an element of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /4\mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /4\mathbb {Z} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9954f452bf8c42d2e57f9fee109cd3b119a9c32d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.072ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /4\mathbb {Z} ,}">⁠ and this element is often denoted by "x mod 4" or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {x}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {x}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/daaa5bc4a6855a8512f663ce36480a3746e202ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.091ex; height:2.676ex;" alt="{\displaystyle {\overline {x}},}"> which is consistent with the notation for 0, 1, 2, 3. The additive inverse of any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {x}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fa4039bbc2a0048c3a3c02e5fd24390cab0dc97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.445ex; height:2.343ex;" alt="{\displaystyle {\overline {x}}}"> in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /4\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /4\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3ded1f832ca4738a18c9e4779381bd9591c058" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /4\mathbb {Z} }">⁠ is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\overline {x}}={\overline {-x}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo>−</mo> <mi>x</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\overline {x}}={\overline {-x}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/012b00df35e4d48800fb2c96e5b4b5c3993d3f65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.251ex; height:2.843ex;" alt="{\displaystyle -{\overline {x}}={\overline {-x}}.}"> For example, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\overline {3}}={\overline {-3}}={\overline {1}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>3</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo>−</mo> <mn>3</mn> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mn>1</mn> <mo accent="false">¯</mo> </mover> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\overline {3}}={\overline {-3}}={\overline {1}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ee7c0f8b89615ff3d2f3ccb13ccc869eb65e8a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.292ex; height:3.009ex;" alt="{\displaystyle -{\overline {3}}={\overline {-3}}={\overline {1}}.}"> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /4\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /4\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3ded1f832ca4738a18c9e4779381bd9591c058" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /4\mathbb {Z} }">⁠ has a subring ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /2\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /2\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b3bb21abe942aa9c0c63bae35a0c38905e1712c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /2\mathbb {Z} }">⁠, and if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"> is prime, then ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /p\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /p\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57869a3a3c4c431cc49c4c7ab1d9c7ea692b517b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.433ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /p\mathbb {Z} }">⁠ has no subrings. <div class="mw-heading mw-heading3"><h3 id="Example:_2-by-2_matrices">Example: 2-by-2 matrices</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=6" title="Edit section: Example: 2-by-2 matrices">edit</a>]</div> The set of 2-by-2 <a href="/wiki/Square_matrices" class="mw-redirect" title="Square matrices">square matrices</a> with entries in a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> F is<a href="#cite_note-FOOTNOTELam2001Theorem_3.1-9">[7]</a><a href="#cite_note-FOOTNOTELang2005Ch_V,_§3-10">[8]</a><a href="#cite_note-FOOTNOTESerre20063-11">[9]</a><a href="#cite_note-FOOTNOTESerre1979158-12">[10]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {M} _{2}(F)=\left\{\left.{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\right|\ a,b,c,d\in F\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>d</mi> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mtext> </mtext> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>d</mi> <mo>∈</mo> <mi>F</mi> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {M} _{2}(F)=\left\{\left.{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\right|\ a,b,c,d\in F\right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/301e5b10afa8978219c607b62983e5040b7fee40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.043ex; height:6.176ex;" alt="{\displaystyle \operatorname {M} _{2}(F)=\left\{\left.{\begin{pmatrix}a&b\\c&d\end{pmatrix}}\right|\ a,b,c,d\in F\right\}.}"></dd></dl> With the operations of matrix addition and <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {M} _{2}(F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {M} _{2}(F)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47e46d4eb6cbb2ef7434911b05cb58bc2ec4ac96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.735ex; height:2.843ex;" alt="{\displaystyle \operatorname {M} _{2}(F)}"> satisfies the above ring axioms. The element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e2c32857c6e3dd650a1af2ce5e3c1af4001c65e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.299ex; height:3.343ex;" alt="{\displaystyle \left({\begin{smallmatrix}1&0\\0&1\end{smallmatrix}}\right)}"> is the multiplicative identity of the ring. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\left({\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\left({\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ac7bfb99cd7b15476b202cc8918d85b98e1e40b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.14ex; height:3.343ex;" alt="{\displaystyle A=\left({\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}\right)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\left({\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\left({\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85adef3f770948fce8dc8439b92f11d5aaa0532d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.195ex; height:3.343ex;" alt="{\displaystyle B=\left({\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}\right),}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB=\left({\begin{smallmatrix}0&0\\0&1\end{smallmatrix}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB=\left({\begin{smallmatrix}0&0\\0&1\end{smallmatrix}}\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e180f61c282e04e3668d3e16f7e232dd2a6970b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.904ex; height:3.343ex;" alt="{\displaystyle AB=\left({\begin{smallmatrix}0&0\\0&1\end{smallmatrix}}\right)}"> while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BA=\left({\begin{smallmatrix}1&0\\0&0\end{smallmatrix}}\right);}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mi>A</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>)</mo> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle BA=\left({\begin{smallmatrix}1&0\\0&0\end{smallmatrix}}\right);}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab21accfa153a9a1b0fc47155bf11b8d1fb644ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.938ex; height:3.343ex;" alt="{\displaystyle BA=\left({\begin{smallmatrix}1&0\\0&0\end{smallmatrix}}\right);}"> this example shows that the ring is noncommutative. More generally, for any ring R, commutative or not, and any nonnegative integer n, the square matrices of dimension n with entries in R form a ring; see <a href="/wiki/Matrix_ring" title="Matrix ring">Matrix ring</a>. <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=7" title="Edit section: History">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Ring_theory#History" title="Ring theory">Ring theory § History</a></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Dedekind.jpeg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Dedekind.jpeg/100px-Dedekind.jpeg" decoding="async" width="100" height="124" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Dedekind.jpeg/150px-Dedekind.jpeg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ca/Dedekind.jpeg/200px-Dedekind.jpeg 2x" data-file-width="262" data-file-height="326" /></a><figcaption><a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a>, one of the founders of <a href="/wiki/Ring_theory" title="Ring theory">ring theory</a></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Dedekind">Dedekind</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=8" title="Edit section: Dedekind">edit</a>]</div> The study of rings originated from the theory of <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial rings</a> and the theory of <a href="/wiki/Algebraic_integer" title="Algebraic integer">algebraic integers</a>.<a href="#cite_note-history-13">[11]</a> In 1871, <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> defined the concept of the ring of integers of a number field.<a href="#cite_note-FOOTNOTEKleiner199827-14">[12]</a> In this context, he introduced the terms "ideal" (inspired by <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Ernst Kummer</a>'s notion of ideal number) and "module" and studied their properties. Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting. <div class="mw-heading mw-heading3"><h3 id="Hilbert">Hilbert</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=9" title="Edit section: Hilbert">edit</a>]</div> The term "Zahlring" (number ring) was coined by <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> in 1892 and published in 1897.<a href="#cite_note-FOOTNOTEHilbert1897-15">[13]</a> In 19th century German, the word "Ring" could mean "association", which is still used today in English in a limited sense (for example, spy ring),[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] so if that were the etymology then it would be similar to the way "group" entered mathematics by being a non-technical word for "collection of related things". According to Harvey Cohn, Hilbert used the term for a ring that had the property of "circling directly back" to an element of itself (in the sense of an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence</a>).<a href="#cite_note-FOOTNOTECohn1980[httpsarchiveorgdetailsadvancednumberth00cohn_0page49_p._49]-16">[14]</a> Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers "cycle back". For instance, if a3 − 4a + 1 = 0 then: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}a^{3}&=4a-1,\\a^{4}&=4a^{2}-a,\\a^{5}&=-a^{2}+16a-4,\\a^{6}&=16a^{2}-8a+1,\\a^{7}&=-8a^{2}+65a-16,\\\vdots \ &\qquad \vdots \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <mi>a</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>4</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mi>a</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>16</mn> <mi>a</mi> <mo>−</mo> <mn>4</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>16</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>8</mn> <mi>a</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>7</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mn>8</mn> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>65</mn> <mi>a</mi> <mo>−</mo> <mn>16</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mspace width="2em" /> <mo>⋮</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}a^{3}&=4a-1,\\a^{4}&=4a^{2}-a,\\a^{5}&=-a^{2}+16a-4,\\a^{6}&=16a^{2}-8a+1,\\a^{7}&=-8a^{2}+65a-16,\\\vdots \ &\qquad \vdots \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c1203276972526f071f434f80d1c18aecec01d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.505ex; width:23.596ex; height:20.176ex;" alt="{\displaystyle {\begin{aligned}a^{3}&=4a-1,\\a^{4}&=4a^{2}-a,\\a^{5}&=-a^{2}+16a-4,\\a^{6}&=16a^{2}-8a+1,\\a^{7}&=-8a^{2}+65a-16,\\\vdots \ &\qquad \vdots \end{aligned}}}"></dd></dl> and so on; in general, an is going to be an integral linear combination of 1, a, and a2. <div class="mw-heading mw-heading3"><h3 id="Fraenkel_and_Noether">Fraenkel and Noether</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=10" title="Edit section: Fraenkel and Noether">edit</a>]</div> The first axiomatic definition of a ring was given by <a href="/wiki/Abraham_Fraenkel" title="Abraham Fraenkel">Adolf Fraenkel</a> in 1915,<a href="#cite_note-FOOTNOTEFraenkel1915143–145-17">[15]</a><a href="#cite_note-FOOTNOTEJacobson200986footnote_1-18">[16]</a> but his axioms were stricter than those in the modern definition. For instance, he required every <a href="/wiki/Zero_divisor" title="Zero divisor">non-zero-divisor</a> to have a <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>.8)_19-0" class="reference"><a href="#cite_note-FOOTNOTEFraenkel1915144axiom_''R''8)-19">[17]</a> In 1921, <a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a> gave a modern axiomatic definition of commutative rings (with and without 1) and developed the foundations of commutative ring theory in her paper Idealtheorie in Ringbereichen.<a href="#cite_note-FOOTNOTENoether192129-20">[18]</a> <div class="mw-heading mw-heading3"><h3 id="Multiplicative_identity_and_the_term_"ring"">Multiplicative identity and the term "ring"</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=11" title="Edit section: Multiplicative identity and the term "ring"">edit</a>]</div> Fraenkel's axioms for a "ring" included that of a multiplicative identity,7)_21-0" class="reference"><a href="#cite_note-FOOTNOTEFraenkel1915144axiom_''R''7)-21">[19]</a> whereas Noether's did not.<a href="#cite_note-FOOTNOTENoether192129-20">[18]</a> Most or all books on algebra<a href="#cite_note-FOOTNOTEvan_der_Waerden1930-22">[20]</a><a href="#cite_note-FOOTNOTEZariskiSamuel1958-23">[21]</a> up to around 1960 followed Noether's convention of not requiring a 1 for a "ring". Starting in the 1960s, it became increasingly common to see books including the existence of 1 in the definition of "ring", especially in advanced books by notable authors such as Artin,<a href="#cite_note-FOOTNOTEArtin2018346-24">[22]</a> Bourbaki,<a href="#cite_note-FOOTNOTEBourbaki198996-25">[23]</a> Eisenbud,<a href="#cite_note-FOOTNOTEEisenbud199511-26">[24]</a> and Lang.<a href="#cite_note-FOOTNOTELang200283-4">[3]</a> There are also books published as late as 2022 that use the term without the requirement for a 1.<a href="#cite_note-FOOTNOTEGallian2006235-27">[25]</a><a href="#cite_note-FOOTNOTEHungerford199742-28">[26]</a><a href="#cite_note-FOOTNOTEWarner1965188-29">[27]</a><a href="#cite_note-FOOTNOTEGarling2022-30">[28]</a> Likewise, the <a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a> does not require unit elements in rings.<a href="#cite_note-31">[29]</a> In a research article, the authors often specify which definition of ring they use in the beginning of that article. Gardner and Wiegandt assert that, when dealing with several objects in the category of rings (as opposed to working with a fixed ring), if one requires all rings to have a 1, then some consequences include the lack of existence of infinite direct sums of rings, and that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable."<a href="#cite_note-FOOTNOTEGardnerWiegandt2003-32">[30]</a> <a href="/wiki/Bjorn_Poonen" title="Bjorn Poonen">Poonen</a> makes the counterargument that the natural notion for rings would be the <a href="/wiki/Direct_product" title="Direct product">direct product</a> rather than the direct sum. However, his main argument is that rings without a multiplicative identity are not totally associative, in the sense that they do not contain the product of any finite sequence of ring elements, including the empty sequence.<a href="#cite_note-33">[c]</a><a href="#cite_note-FOOTNOTEPoonen2019-34">[31]</a> Authors who follow either convention for the use of the term "ring" may use one of the following terms to refer to objects satisfying the other convention: <dl><dd><ul><li>to include a requirement for a multiplicative identity: "unital ring", "unitary ring", "unit ring", "ring with unity", "ring with identity", "ring with a unit",<a href="#cite_note-FOOTNOTEWilder1965176-35">[32]</a> or "ring with 1".<a href="#cite_note-FOOTNOTERotman19987-36">[33]</a></li> <li>to omit a requirement for a multiplicative identity: "rng"<a href="#cite_note-FOOTNOTEJacobson2009155-37">[34]</a> or "pseudo-ring",<a href="#cite_note-FOOTNOTEBourbaki198998-38">[35]</a> although the latter may be confusing because it also has other meanings.</li></ul></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Basic_examples">Basic examples</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=12" title="Edit section: Basic examples">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Associative_algebra#Examples" title="Associative algebra">Associative algebra § Examples</a></div> <div class="mw-heading mw-heading3"><h3 id="Commutative_rings">Commutative rings</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=13" title="Edit section: Commutative rings">edit</a>]</div> <ul><li>The prototypical example is the ring of integers with the two operations of addition and multiplication.</li> <li>The rational, real and complex numbers are commutative rings of a type called <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>.</li> <li>A unital associative <a href="/wiki/Algebra_over_a_ring" class="mw-redirect" title="Algebra over a ring">algebra over a commutative ring</a> R is itself a ring as well as an <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">R-module</a>. Some examples: <ul><li>The algebra R[X] of <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomials</a> with coefficients in R.</li> <li>The algebra <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[[X_{1},\dots ,X_{n}]]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mo stretchy="false">[</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[[X_{1},\dots ,X_{n}]]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9faeec0b7c5fd0fcd5c02b2d94988eb501c67591" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.651ex; height:2.843ex;" alt="{\displaystyle R[[X_{1},\dots ,X_{n}]]}"> of <a href="/wiki/Formal_power_series" title="Formal power series">formal power series</a> with coefficients in R.</li> <li>The set of all <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> real-valued <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> defined on the real line forms a commutative ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">⁠-algebra. The operations are <a href="/wiki/Pointwise" title="Pointwise">pointwise</a> addition and multiplication of functions.</li> <li>Let X be a set, and let R be a ring. Then the set of all functions from X to R forms a ring, which is commutative if R is commutative.</li></ul></li> <li>The ring of <a href="/wiki/Quadratic_integers" class="mw-redirect" title="Quadratic integers">quadratic integers</a>, the integral closure of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">⁠ in a quadratic extension of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869719f08f506bf866043442858fb3da1d4b4b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} .}">⁠ It is a subring of the ring of all <a href="/wiki/Algebraic_integers" class="mw-redirect" title="Algebraic integers">algebraic integers</a>.</li> <li>The ring of <a href="/wiki/Profinite_integer" title="Profinite integer">profinite integers</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\widehat {\mathbb {Z} }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\widehat {\mathbb {Z} }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8098caf3ca92d1ff05ae06a57d6c0f8f074939c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.197ex; height:3.176ex;" alt="{\displaystyle {\widehat {\mathbb {Z} }},}">⁠ the (infinite) product of the rings of p-adic integers ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}">⁠ over all prime numbers p.</li> <li>The <a href="/wiki/Hecke_algebra" title="Hecke algebra">Hecke ring</a>, the ring generated by Hecke operators.</li> <li>If S is a set, then the <a href="/wiki/Power_set" title="Power set">power set</a> of S becomes a ring if we define addition to be the <a href="/wiki/Symmetric_difference" title="Symmetric difference">symmetric difference</a> of sets and multiplication to be <a href="/wiki/Intersection_(set_theory)" title="Intersection (set theory)">intersection</a>. This is an example of a <a href="/wiki/Boolean_ring" title="Boolean ring">Boolean ring</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Noncommutative_rings">Noncommutative rings</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=14" title="Edit section: Noncommutative rings">edit</a>]</div> <ul><li>For any ring R and any natural number n, the set of all square n-by-n <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> with entries from R, forms a ring with matrix addition and matrix multiplication as operations. For n = 1, this matrix ring is isomorphic to R itself. For n > 1 (and R not the zero ring), this matrix ring is noncommutative.</li> <li>If G is an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a>, then the <a href="/wiki/Group_homomorphism" title="Group homomorphism">endomorphisms</a> of G form a ring, the <a href="/wiki/Endomorphism_ring" title="Endomorphism ring">endomorphism ring</a> End(G) of G. The operations in this ring are addition and composition of endomorphisms. More generally, if V is a <a href="/wiki/Left_module" class="mw-redirect" title="Left module">left module</a> over a ring R, then the set of all R-linear maps forms a ring, also called the endomorphism ring and denoted by EndR(V).</li> <li>The <a href="/wiki/Endomorphism_ring_of_an_elliptic_curve" class="mw-redirect" title="Endomorphism ring of an elliptic curve">endomorphism ring of an elliptic curve</a>. It is a commutative ring if the elliptic curve is defined over a field of characteristic zero.</li> <li>If G is a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> and R is a ring, the <a href="/wiki/Group_ring" title="Group ring">group ring</a> of G over R is a <a href="/wiki/Free_module" title="Free module">free module</a> over R having G as basis. Multiplication is defined by the rules that the elements of G commute with the elements of R and multiply together as they do in the group G.</li> <li>The <a href="/wiki/Ring_of_differential_operators" class="mw-redirect" title="Ring of differential operators">ring of differential operators</a> (depending on the context). In fact, many rings that appear in analysis are noncommutative. For example, most <a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebras</a> are noncommutative.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Non-rings">Non-rings</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=15" title="Edit section: Non-rings">edit</a>]</div> <ul><li>The set of <a href="/wiki/Natural_number" title="Natural number">natural numbers</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }">⁠ with the usual operations is not a ring, since ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\mathbb {N} ,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbb {N} ,+)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0072a6ee0ab943ce24dc44083bd60d50739a0b1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle (\mathbb {N} ,+)}">⁠ is not even a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> (not all the elements are <a href="/wiki/Inverse_element" title="Inverse element">invertible</a> with respect to addition – for instance, there is no natural number which can be added to 3 to get 0 as a result). There is a natural way to enlarge it to a ring, by including negative numbers to produce the ring of integers ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89f4f38f32c2068bca9dc701d13b03dd4a5d52ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} .}">⁠ The natural numbers (including 0) form an algebraic structure known as a <a href="/wiki/Semiring" title="Semiring">semiring</a> (which has all of the axioms of a ring excluding that of an additive inverse).</li> <li>Let R be the set of all continuous functions on the real line that vanish outside a bounded interval that depends on the function, with addition as usual but with multiplication defined as <a href="/wiki/Convolution" title="Convolution">convolution</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(x)=\int _{-\infty }^{\infty }f(y)g(x-y)\,dy.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)(x)=\int _{-\infty }^{\infty }f(y)g(x-y)\,dy.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/294affe71f6e28d18165e12aac5f9bad340d8558" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.369ex; height:6.009ex;" alt="{\displaystyle (f*g)(x)=\int _{-\infty }^{\infty }f(y)g(x-y)\,dy.}"> Then R is a rng, but not a ring: the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a> has the property of a multiplicative identity, but it is not a function and hence is not an element of R.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Basic_concepts">Basic concepts</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=16" title="Edit section: Basic concepts">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Products_and_powers">Products and powers</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=17" title="Edit section: Products and powers">edit</a>]</div> For each nonnegative integer n, given a sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a_{1},\dots ,a_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a_{1},\dots ,a_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f32ea8be17cff0b2d07309b931aa8a6af949ac8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.72ex; height:2.843ex;" alt="{\displaystyle (a_{1},\dots ,a_{n})}"> of n elements of R, one can define the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{n}=\prod _{i=1}^{n}a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{n}=\prod _{i=1}^{n}a_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6dc7ea3e7d5fa04def71ebe5c042a67804d00bbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.195ex; height:6.843ex;" alt="{\displaystyle P_{n}=\prod _{i=1}^{n}a_{i}}"> recursively: let P0 = 1 and let Pm = Pm−1am for 1 ≤ m ≤ n. As a special case, one can define nonnegative integer powers of an element a of a ring: a0 = 1 and an = an−1a for n ≥ 1. Then am+n = aman for all m, n ≥ 0. <div class="mw-heading mw-heading3"><h3 id="Elements_in_a_ring">Elements in a ring</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=18" title="Edit section: Elements in a ring">edit</a>]</div> A left <a href="/wiki/Zero_divisor" title="Zero divisor">zero divisor</a> of a ring R is an element a in the ring such that there exists a nonzero element b of R such that ab = 0.<a href="#cite_note-39">[d]</a> A right zero divisor is defined similarly. A <a href="/wiki/Nilpotent_element" class="mw-redirect" title="Nilpotent element">nilpotent element</a> is an element a such that an = 0 for some n > 0. One example of a nilpotent element is a <a href="/wiki/Nilpotent_matrix" title="Nilpotent matrix">nilpotent matrix</a>. A nilpotent element in a <a href="/wiki/Zero_ring" title="Zero ring">nonzero ring</a> is necessarily a zero divisor. An <a href="/wiki/Idempotent_element_(ring_theory)" class="mw-redirect" title="Idempotent element (ring theory)">idempotent</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>e</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd253103f0876afc68ebead27a5aa9867d927467" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}"> is an element such that e2 = e. One example of an idempotent element is a <a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">projection</a> in linear algebra. A <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit</a> is an element a having a <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>; in this case the inverse is unique, and is denoted by a–1. The set of units of a ring is a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> under ring multiplication; this group is denoted by R× or R* or U(R). For example, if R is the ring of all square matrices of size n over a field, then R× consists of the set of all invertible matrices of size n, and is called the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a>. <div class="mw-heading mw-heading3"><h3 id="Subring">Subring</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=19" title="Edit section: Subring">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Subring" title="Subring">Subring</a></div> A subset S of R is called a <a href="/wiki/Subring" title="Subring">subring</a> if any one of the following equivalent conditions holds: <ul><li>the addition and multiplication of R <a href="/wiki/Restricted_function" class="mw-redirect" title="Restricted function">restrict</a> to give operations S × S → S making S a ring with the same multiplicative identity as R.</li> <li>1 ∈ S; and for all x, y in S, the elements xy, x + y, and −x are in S.</li> <li>S can be equipped with operations making it a ring such that the inclusion map S → R is a ring homomorphism.</li></ul> For example, the ring ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">⁠ of integers is a subring of the <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> of real numbers and also a subring of the ring of <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [X]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a538d203a057d4c604f799c28e9a7be410fdcac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.824ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [X]}">⁠ (in both cases, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">⁠ contains 1, which is the multiplicative identity of the larger rings). On the other hand, the subset of even integers ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c112372c16743e8c02db5eac5c6de0659841bec6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle 2\mathbb {Z} }">⁠ does not contain the identity element 1 and thus does not qualify as a subring of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/237089e243508123c48781cce997626d9c012e7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.197ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ;}">⁠ one could call ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c112372c16743e8c02db5eac5c6de0659841bec6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle 2\mathbb {Z} }">⁠ a <a href="/wiki/Rng_(algebra)" title="Rng (algebra)">subrng</a>, however. An intersection of subrings is a subring. Given a subset E of R, the smallest subring of R containing E is the intersection of all subrings of R containing E, and it is called the subring generated by E. For a ring R, the smallest subring of R is called the characteristic subring of R. It can be generated through addition of copies of 1 and −1. It is possible that n · 1 = 1 + 1 + ... + 1 (n times) can be zero. If n is the smallest positive integer such that this occurs, then n is called the <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> of R. In some rings, n · 1 is never zero for any positive integer n, and those rings are said to have characteristic zero. Given a ring R, let Z(R) denote the set of all elements x in R such that x commutes with every element in R: xy = yx for any y in R. Then Z(R) is a subring of R, called the <a href="/wiki/Center_(ring_theory)" title="Center (ring theory)">center</a> of R. More generally, given a subset X of R, let S be the set of all elements in R that commute with every element in X. Then S is a subring of R, called the <a href="/wiki/Centralizer_(ring_theory)" class="mw-redirect" title="Centralizer (ring theory)">centralizer</a> (or commutant) of X. The center is the centralizer of the entire ring R. Elements or subsets of the center are said to be central in R; they (each individually) generate a subring of the center. <div class="mw-heading mw-heading3"><h3 id="Ideal">Ideal</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=20" title="Edit section: Ideal">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">Ideal (ring theory)</a></div> Let R be a ring. A left ideal of R is a nonempty subset I of R such that for any x, y in I and r in R, the elements x + y and rx are in I. If R I denotes the R-span of I, that is, the set of finite sums <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{1}x_{1}+\cdots +r_{n}x_{n}\quad {\textrm {such}}\;{\textrm {that}}\;r_{i}\in R\;{\textrm {and}}\;x_{i}\in I,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>such</mtext> </mrow> </mrow> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>that</mtext> </mrow> </mrow> <mspace width="thickmathspace" /> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>R</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mrow> <mspace width="thickmathspace" /> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>I</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{1}x_{1}+\cdots +r_{n}x_{n}\quad {\textrm {such}}\;{\textrm {that}}\;r_{i}\in R\;{\textrm {and}}\;x_{i}\in I,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/828ef119c361f6abf95b844ad6f557580d84f7b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:48.396ex; height:2.509ex;" alt="{\displaystyle r_{1}x_{1}+\cdots +r_{n}x_{n}\quad {\textrm {such}}\;{\textrm {that}}\;r_{i}\in R\;{\textrm {and}}\;x_{i}\in I,}"></dd></dl> then I is a left ideal if RI ⊆ I. Similarly, a right ideal is a subset I such that IR ⊆ I. A subset I is said to be a two-sided ideal or simply ideal if it is both a left ideal and right ideal. A one-sided or two-sided ideal is then an additive subgroup of R. If E is a subset of R, then RE is a left ideal, called the left ideal generated by E; it is the smallest left ideal containing E. Similarly, one can consider the right ideal or the two-sided ideal generated by a subset of R. If x is in R, then Rx and xR are left ideals and right ideals, respectively; they are called the <a href="/wiki/Principal_ideal" title="Principal ideal">principal</a> left ideals and right ideals generated by x. The principal ideal RxR is written as (x). For example, the set of all positive and negative multiples of 2 along with 0 form an ideal of the integers, and this ideal is generated by the integer 2. In fact, every ideal of the ring of integers is principal. Like a group, a ring is said to be <a href="/wiki/Simple_ring" title="Simple ring">simple</a> if it is nonzero and it has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field. Rings are often studied with special conditions set upon their ideals. For example, a ring in which there is no strictly increasing infinite <a href="/wiki/Total_order#Chains" title="Total order">chain</a> of left ideals is called a left <a href="/wiki/Noetherian_ring" title="Noetherian ring">Noetherian ring</a>. A ring in which there is no strictly decreasing infinite chain of left ideals is called a left <a href="/wiki/Artinian_ring" title="Artinian ring">Artinian ring</a>. It is a somewhat surprising fact that a left Artinian ring is left Noetherian (the <a href="/wiki/Hopkins%E2%80%93Levitzki_theorem" title="Hopkins–Levitzki theorem">Hopkins–Levitzki theorem</a>). The integers, however, form a Noetherian ring which is not Artinian. For commutative rings, the ideals generalize the classical notion of divisibility and decomposition of an integer into prime numbers in algebra. A proper ideal P of R is called a <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a> if for any elements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd941136d77e1d2a54ec7a326c4216d9e2ffa587" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.124ex; height:2.509ex;" alt="{\displaystyle x,y\in R}"> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xy\in P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>y</mi> <mo>∈</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xy\in P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/298c1a56c15cf0821836a96909e59aa15ee4a067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.071ex; height:2.509ex;" alt="{\displaystyle xy\in P}"> implies either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cc98d5d7d5ae8afe28ea054450cac2af7bca3d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.916ex; height:2.176ex;" alt="{\displaystyle x\in P}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in P.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>P</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in P.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19f0998dac9af00ca54173db6d48fff7cf185128" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.388ex; height:2.509ex;" alt="{\displaystyle y\in P.}"> Equivalently, P is prime if for any ideals I, J we have that IJ ⊆ P implies either I ⊆ P or J ⊆ P. This latter formulation illustrates the idea of ideals as generalizations of elements. <div class="mw-heading mw-heading3"><h3 id="Homomorphism">Homomorphism</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=21" title="Edit section: Homomorphism">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">Ring homomorphism</a></div> A <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">homomorphism</a> from a ring (R, +, ⋅) to a ring (S, ‡, ∗) is a function f from R to S that preserves the ring operations; namely, such that, for all a, b in R the following identities hold: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&f(a+b)=f(a)\ddagger f(b)\\&f(a\cdot b)=f(a)*f(b)\\&f(1_{R})=1_{S}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>‡</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&f(a+b)=f(a)\ddagger f(b)\\&f(a\cdot b)=f(a)*f(b)\\&f(1_{R})=1_{S}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2553fe96c6adf91c05410aecb1a18505a42441cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.473ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}&f(a+b)=f(a)\ddagger f(b)\\&f(a\cdot b)=f(a)*f(b)\\&f(1_{R})=1_{S}\end{aligned}}}"></dd></dl> If one is working with rngs, then the third condition is dropped. A ring homomorphism f is said to be an <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> if there exists an inverse homomorphism to f (that is, a ring homomorphism that is an <a href="/wiki/Inverse_function" title="Inverse function">inverse function</a>), or equivalently if it is <a href="/wiki/Bijection" title="Bijection">bijective</a>. Examples: <ul><li>The function that maps each integer x to its remainder modulo 4 (a number in {0, 1, 2, 3}) is a homomorphism from the ring ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">⁠ to the quotient ring ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} /4\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>4</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} /4\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3ded1f832ca4738a18c9e4779381bd9591c058" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.426ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /4\mathbb {Z} }">⁠ ("quotient ring" is defined below).</li> <li>If u is a unit element in a ring R, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\to R,x\mapsto uxu^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">→</mo> <mi>R</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">↦</mo> <mi>u</mi> <mi>x</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\to R,x\mapsto uxu^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb810ddb07e05bc04b530a55e529bc372ab924ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.442ex; height:3.009ex;" alt="{\displaystyle R\to R,x\mapsto uxu^{-1}}"> is a ring homomorphism, called an <a href="/wiki/Inner_automorphism" title="Inner automorphism">inner automorphism</a> of R.</li> <li>Let R be a commutative ring of prime characteristic p. Then x ↦ xp is a ring endomorphism of R called the <a href="/wiki/Frobenius_homomorphism" class="mw-redirect" title="Frobenius homomorphism">Frobenius homomorphism</a>.</li> <li>The <a href="/wiki/Galois_group" title="Galois group">Galois group</a> of a field extension L / K is the set of all automorphisms of L whose restrictions to K are the identity.</li> <li>For any ring R, there are a unique ring homomorphism ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} \mapsto R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">↦</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \mapsto R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1254ca3029e362f5a586691b6cf2896de1d6e1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.928ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} \mapsto R}">⁠ and a unique ring homomorphism R → 0.</li> <li>An <a href="/wiki/Epimorphism" title="Epimorphism">epimorphism</a> (that is, right-cancelable morphism) of rings need not be surjective. For example, the unique map ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} \to \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \to \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fcf77268bcc36d0149c3388c50868effdd7f417" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.973ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} \to \mathbb {Q} }">⁠ is an epimorphism.</li> <li>An algebra homomorphism from a k-algebra to the <a href="/wiki/Endomorphism_algebra" class="mw-redirect" title="Endomorphism algebra">endomorphism algebra</a> of a vector space over k is called a <a href="/wiki/Algebra_representation" title="Algebra representation">representation of the algebra</a>.</li></ul> Given a ring homomorphism f : R → S, the set of all elements mapped to 0 by f is called the <a href="/wiki/Kernel_of_a_ring_homomorphism" class="mw-redirect" title="Kernel of a ring homomorphism">kernel</a> of f. The kernel is a two-sided ideal of R. The image of f, on the other hand, is not always an ideal, but it is always a subring of S. To give a ring homomorphism from a commutative ring R to a ring A with image contained in the center of A is the same as to give a structure of an <a href="/wiki/Associative_algebra" title="Associative algebra">algebra</a> over R to A (which in particular gives a structure of an A-module). <div class="mw-heading mw-heading3"><h3 id="Quotient_ring">Quotient ring</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=22" title="Edit section: Quotient ring">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Quotient_ring" title="Quotient ring">Quotient ring</a></div> The notion of <a href="/wiki/Quotient_ring" title="Quotient ring">quotient ring</a> is analogous to the notion of a <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a>. Given a ring (R, +, ⋅) and a two-sided <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> I of (R, +, ⋅), view I as subgroup of (R, +); then the quotient ring R / I is the set of <a href="/wiki/Coset" title="Coset">cosets</a> of I together with the operations <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&(a+I)+(b+I)=(a+b)+I,\\&(a+I)(b+I)=(ab)+I.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>I</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>I</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&(a+I)+(b+I)=(a+b)+I,\\&(a+I)(b+I)=(ab)+I.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/613200ac47a306f8080e74b8f71f82bc48dfa4ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.097ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&(a+I)+(b+I)=(a+b)+I,\\&(a+I)(b+I)=(ab)+I.\end{aligned}}}"></dd></dl> for all a, b in R. The ring R / I is also called a factor ring. As with a quotient group, there is a canonical homomorphism p : R → R / I, given by x ↦ x + I. It is surjective and satisfies the following universal property: <ul><li>If f : R → S is a ring homomorphism such that f(I) = 0, then there is a unique homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {f}}:R/I\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>:</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>I</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {f}}:R/I\to S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa9d21aa213614fce915af61d2d38cb9574a37d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.633ex; height:3.509ex;" alt="{\displaystyle {\overline {f}}:R/I\to S}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f={\overline {f}}\circ p.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∘</mo> <mi>p</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f={\overline {f}}\circ p.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e294a5e001dbfc1c5669393f066359a98b99f4b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.872ex; height:3.343ex;" alt="{\displaystyle f={\overline {f}}\circ p.}"></li></ul> For any ring homomorphism f : R → S, invoking the universal property with I = ker f produces a homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {f}}:R/\ker f\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>:</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ker</mi> <mo>⁡</mo> <mi>f</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {f}}:R/\ker f\to S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5497d1fb5d13fabfa8476a47457f152d6165ff30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.685ex; height:3.509ex;" alt="{\displaystyle {\overline {f}}:R/\ker f\to S}"> that gives an isomorphism from R / ker f to the image of f. <div class="mw-heading mw-heading2"><h2 id="Module">Module</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=23" title="Edit section: Module">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module (mathematics)</a></div> The concept of a module over a ring generalizes the concept of a <a href="/wiki/Vector_space" title="Vector space">vector space</a> (over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>) by generalizing from multiplication of vectors with elements of a field (<a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a>) to multiplication with elements of a ring. More precisely, given a ring R, an R-module M is an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> equipped with an <a href="/wiki/Operation_(mathematics)" title="Operation (mathematics)">operation</a> R × M → M (associating an element of M to every pair of an element of R and an element of M) that satisfies certain <a href="/wiki/Axiom#Non-logical_axioms" title="Axiom">axioms</a>. This operation is commonly denoted by juxtaposition and called multiplication. The axioms of modules are the following: for all a, b in R and all x, y in M, <dl><dd>M is an abelian group under addition.</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&a(x+y)=ax+ay\\&(a+b)x=ax+bx\\&1x=x\\&(ab)x=a(bx)\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>a</mi> <mi>y</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mi>x</mi> <mo>+</mo> <mi>b</mi> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mn>1</mn> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>=</mo> <mi>a</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&a(x+y)=ax+ay\\&(a+b)x=ax+bx\\&1x=x\\&(ab)x=a(bx)\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32dd53b7bbb701685499b47335200650201d401c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:20ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}&a(x+y)=ax+ay\\&(a+b)x=ax+bx\\&1x=x\\&(ab)x=a(bx)\end{aligned}}}"></dd></dl> When the ring is <a href="/wiki/Noncommutative_ring" title="Noncommutative ring">noncommutative</a> these axioms define left modules; right modules are defined similarly by writing xa instead of ax. This is not only a change of notation, as the last axiom of right modules (that is x(ab) = (xa)b) becomes (ab)x = b(ax), if left multiplication (by ring elements) is used for a right module. Basic examples of modules are ideals, including the ring itself. Although similarly defined, the theory of modules is much more complicated than that of vector space, mainly, because, unlike vector spaces, modules are not characterized (up to an isomorphism) by a single invariant (the <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">dimension of a vector space</a>). In particular, not all modules have a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a>. The axioms of modules imply that (−1)x = −x, where the first minus denotes the <a href="/wiki/Additive_inverse" title="Additive inverse">additive inverse</a> in the ring and the second minus the additive inverse in the module. Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers. Any ring homomorphism induces a structure of a module: if f : R → S is a ring homomorphism, then S is a left module over R by the multiplication: rs = f(r)s. If R is commutative or if f(R) is contained in the <a href="/wiki/Center_of_a_ring" class="mw-redirect" title="Center of a ring">center</a> of S, the ring S is called a R-<a href="/wiki/Algebra_over_a_ring" class="mw-redirect" title="Algebra over a ring">algebra</a>. In particular, every ring is an algebra over the integers. <div class="mw-heading mw-heading2"><h2 id="Constructions">Constructions</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=24" title="Edit section: Constructions">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Direct_product">Direct product</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=25" title="Edit section: Direct product">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Direct_product_of_rings" class="mw-redirect" title="Direct product of rings">Direct product of rings</a></div> Let R and S be rings. Then the <a href="/wiki/Cartesian_product" title="Cartesian product">product</a> R × S can be equipped with the following natural ring structure: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&(r_{1},s_{1})+(r_{2},s_{2})=(r_{1}+r_{2},s_{1}+s_{2})\\&(r_{1},s_{1})\cdot (r_{2},s_{2})=(r_{1}\cdot r_{2},s_{1}\cdot s_{2})\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⋅</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&(r_{1},s_{1})+(r_{2},s_{2})=(r_{1}+r_{2},s_{1}+s_{2})\\&(r_{1},s_{1})\cdot (r_{2},s_{2})=(r_{1}\cdot r_{2},s_{1}\cdot s_{2})\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67b317bd44af52eace2fcb352bf7c7651cb9ef41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:37.891ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&(r_{1},s_{1})+(r_{2},s_{2})=(r_{1}+r_{2},s_{1}+s_{2})\\&(r_{1},s_{1})\cdot (r_{2},s_{2})=(r_{1}\cdot r_{2},s_{1}\cdot s_{2})\end{aligned}}}"></dd></dl> for all r1, r2 in R and s1, s2 in S. The ring R × S with the above operations of addition and multiplication and the multiplicative identity (1, 1) is called the <a href="/wiki/Direct_product_of_rings" class="mw-redirect" title="Direct product of rings">direct product</a> of R with S. The same construction also works for an arbitrary family of rings: if Ri are rings indexed by a set I, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \prod _{i\in I}R_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>I</mi> </mrow> </munder> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \prod _{i\in I}R_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3925beb567895538aa9c1945dfa914536eeab778" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.869ex; height:3.009ex;" alt="{\textstyle \prod _{i\in I}R_{i}}"> is a ring with componentwise addition and multiplication. Let R be a commutative ring and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{1},\cdots ,{\mathfrak {a}}_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{1},\cdots ,{\mathfrak {a}}_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9454ea2b80c229917fa96121807f90e4476ec6b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.776ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {a}}_{1},\cdots ,{\mathfrak {a}}_{n}}"> be ideals such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{i}+{\mathfrak {a}}_{j}=(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{i}+{\mathfrak {a}}_{j}=(1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57c9f995e0b4e3bc8c7bfd08e33d624bc973ef39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.945ex; height:3.009ex;" alt="{\displaystyle {\mathfrak {a}}_{i}+{\mathfrak {a}}_{j}=(1)}"> whenever i ≠ j. Then the <a href="/wiki/Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a> says there is a canonical ring isomorphism: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R/{\textstyle \bigcap _{i=1}^{n}{{\mathfrak {a}}_{i}}}\simeq \prod _{i=1}^{n}{R/{\mathfrak {a}}_{i}},\qquad x{\bmod {\textstyle \bigcap _{i=1}^{n}{\mathfrak {a}}_{i}}}\mapsto (x{\bmod {\mathfrak {a}}}_{1},\ldots ,x{\bmod {\mathfrak {a}}}_{n}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mstyle> </mrow> <mo>≃</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> </mrow> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>x</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R/{\textstyle \bigcap _{i=1}^{n}{{\mathfrak {a}}_{i}}}\simeq \prod _{i=1}^{n}{R/{\mathfrak {a}}_{i}},\qquad x{\bmod {\textstyle \bigcap _{i=1}^{n}{\mathfrak {a}}_{i}}}\mapsto (x{\bmod {\mathfrak {a}}}_{1},\ldots ,x{\bmod {\mathfrak {a}}}_{n}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c93a1164449f48b76d0b05784ffcfa7607839685" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:71.198ex; height:6.843ex;" alt="{\displaystyle R/{\textstyle \bigcap _{i=1}^{n}{{\mathfrak {a}}_{i}}}\simeq \prod _{i=1}^{n}{R/{\mathfrak {a}}_{i}},\qquad x{\bmod {\textstyle \bigcap _{i=1}^{n}{\mathfrak {a}}_{i}}}\mapsto (x{\bmod {\mathfrak {a}}}_{1},\ldots ,x{\bmod {\mathfrak {a}}}_{n}).}"> A "finite" direct product may also be viewed as a direct sum of ideals.<a href="#cite_note-FOOTNOTECohn2003Theorem_4.5.1-40">[36]</a> Namely, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{i},1\leq i\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{i},1\leq i\leq n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/371858f9c96f8351218374fa0a81a543509b37a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.154ex; height:2.509ex;" alt="{\displaystyle R_{i},1\leq i\leq n}"> be rings, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle R_{i}\to R=\prod R_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">→</mo> <mi>R</mi> <mo>=</mo> <mo>∏</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle R_{i}\to R=\prod R_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbed16de8bf260389bf86cc1a70476b7224e61be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.185ex; height:2.843ex;" alt="{\textstyle R_{i}\to R=\prod R_{i}}"> the inclusions with the images <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e599f543e7b9df295f4407630f1a1123b8ac90e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.962ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {a}}_{i}}"> (in particular <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e599f543e7b9df295f4407630f1a1123b8ac90e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.962ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {a}}_{i}}"> are rings though not subrings). Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e599f543e7b9df295f4407630f1a1123b8ac90e2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.962ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {a}}_{i}}"> are ideals of R and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R={\mathfrak {a}}_{1}\oplus \cdots \oplus {\mathfrak {a}}_{n},\quad {\mathfrak {a}}_{i}{\mathfrak {a}}_{j}=0,i\neq j,\quad {\mathfrak {a}}_{i}^{2}\subseteq {\mathfrak {a}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊕</mo> <mo>⋯</mo> <mo>⊕</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>i</mi> <mo>≠</mo> <mi>j</mi> <mo>,</mo> <mspace width="1em" /> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>⊆</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R={\mathfrak {a}}_{1}\oplus \cdots \oplus {\mathfrak {a}}_{n},\quad {\mathfrak {a}}_{i}{\mathfrak {a}}_{j}=0,i\neq j,\quad {\mathfrak {a}}_{i}^{2}\subseteq {\mathfrak {a}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c726747076bb3a1bf1f5416790d13354b2866748" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.043ex; height:3.176ex;" alt="{\displaystyle R={\mathfrak {a}}_{1}\oplus \cdots \oplus {\mathfrak {a}}_{n},\quad {\mathfrak {a}}_{i}{\mathfrak {a}}_{j}=0,i\neq j,\quad {\mathfrak {a}}_{i}^{2}\subseteq {\mathfrak {a}}_{i}}"> as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums). Clearly the direct sum of such ideals also defines a product of rings that is isomorphic to R. Equivalently, the above can be done through <a href="/wiki/Central_idempotent" class="mw-redirect" title="Central idempotent">central idempotents</a>. Assume that R has the above decomposition. Then we can write <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=e_{1}+\cdots +e_{n},\quad e_{i}\in {\mathfrak {a}}_{i}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>=</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1=e_{1}+\cdots +e_{n},\quad e_{i}\in {\mathfrak {a}}_{i}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24447f71ff15b9acb2c1f520137712f84376f1a2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:27.794ex; height:2.509ex;" alt="{\displaystyle 1=e_{1}+\cdots +e_{n},\quad e_{i}\in {\mathfrak {a}}_{i}.}"> By the conditions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5d44ffe7b575bec4c3e2e7230786c2cc568956d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.609ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {a}}_{i},}"> one has that ei are central idempotents and eiej = 0, i ≠ j (orthogonal). Again, one can reverse the construction. Namely, if one is given a partition of 1 in orthogonal central idempotents, then let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{i}=Re_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>R</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{i}=Re_{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/040295c2e18e3158e8b397b4abcb7e74e5adff12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.355ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {a}}_{i}=Re_{i},}"> which are two-sided ideals. If each ei is not a sum of orthogonal central idempotents,<a href="#cite_note-41">[e]</a> then their direct sum is isomorphic to R. An important application of an infinite direct product is the construction of a <a href="/wiki/Projective_limit" class="mw-redirect" title="Projective limit">projective limit</a> of rings (see below). Another application is a <a href="/wiki/Restricted_product" title="Restricted product">restricted product</a> of a family of rings (cf. <a href="/wiki/Adele_ring" title="Adele ring">adele ring</a>). <div class="mw-heading mw-heading3"><h3 id="Polynomial_ring">Polynomial ring</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=26" title="Edit section: Polynomial ring">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Polynomial_ring" title="Polynomial ring">Polynomial ring</a></div> Given a symbol t (called a variable) and a commutative ring R, the set of polynomials <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[t]=\left\{a_{n}t^{n}+a_{n-1}t^{n-1}+\dots +a_{1}t+a_{0}\mid n\geq 0,a_{j}\in R\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>t</mi> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>∣</mo> <mi>n</mi> <mo>≥</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>∈</mo> <mi>R</mi> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[t]=\left\{a_{n}t^{n}+a_{n-1}t^{n-1}+\dots +a_{1}t+a_{0}\mid n\geq 0,a_{j}\in R\right\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eebb362f161716c1bcb50a5656fbfbc21532a5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:57.783ex; height:3.343ex;" alt="{\displaystyle R[t]=\left\{a_{n}t^{n}+a_{n-1}t^{n-1}+\dots +a_{1}t+a_{0}\mid n\geq 0,a_{j}\in R\right\}}"></dd></dl> forms a commutative ring with the usual addition and multiplication, containing R as a subring. It is called the <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial ring</a> over R. More generally, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\left[t_{1},\ldots ,t_{n}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow> <mo>[</mo> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\left[t_{1},\ldots ,t_{n}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc4cf3e812fcb5095239260dbc2ba56ceae257e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.575ex; height:2.843ex;" alt="{\displaystyle R\left[t_{1},\ldots ,t_{n}\right]}"> of all polynomials in variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{1},\ldots ,t_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{1},\ldots ,t_{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d5d47a886224ab3923f286622880208346f172a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.13ex; height:2.343ex;" alt="{\displaystyle t_{1},\ldots ,t_{n}}"> forms a commutative ring, containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\left[t_{i}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow> <mo>[</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\left[t_{i}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e9b4f740e9cf30c4b29d9510608e04a28906ec6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.084ex; height:2.843ex;" alt="{\displaystyle R\left[t_{i}\right]}"> as subrings. If R is an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>, then R[t] is also an integral domain; its field of fractions is the field of <a href="/wiki/Rational_function" title="Rational function">rational functions</a>. If R is a Noetherian ring, then R[t] is a Noetherian ring. If R is a unique factorization domain, then R[t] is a unique factorization domain. Finally, R is a field if and only if R[t] is a principal ideal domain. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\subseteq S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>⊆</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\subseteq S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61ab53336ab789637a720d5ea2ed48097cbf9878" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.362ex; height:2.343ex;" alt="{\displaystyle R\subseteq S}"> be commutative rings. Given an element x of S, one can consider the ring homomorphism <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[t]\to S,\quad f\mapsto f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo>,</mo> <mspace width="1em" /> <mi>f</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[t]\to S,\quad f\mapsto f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0226daa10f64386e959d3aadf12fb73e82b45082" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.677ex; height:2.843ex;" alt="{\displaystyle R[t]\to S,\quad f\mapsto f(x)}"></dd></dl> (that is, the <a href="/wiki/Substitution_(algebra)" class="mw-redirect" title="Substitution (algebra)">substitution</a>). If S = R[t] and x = t, then f(t) = f. Because of this, the polynomial f is often also denoted by f(t). The image of the map ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\mapsto f(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\mapsto f(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea98952f3899f06730b085ac6328fcce26293152" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.31ex; height:2.843ex;" alt="{\displaystyle f\mapsto f(x)}">⁠ is denoted by R[x]; it is the same thing as the subring of S generated by R and x. Example: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\left[t^{2},t^{3}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mrow> <mo>[</mo> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\left[t^{2},t^{3}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e4541532126a442f0b3099109c17c60a30b0bb5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.359ex; height:3.343ex;" alt="{\displaystyle k\left[t^{2},t^{3}\right]}"> denotes the image of the homomorphism <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[x,y]\to k[t],\,f\mapsto f\left(t^{2},t^{3}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi>k</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mi>f</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k[x,y]\to k[t],\,f\mapsto f\left(t^{2},t^{3}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70de42b23023914ef0b1a1e71bcff1fdc76af67b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:28.947ex; height:3.343ex;" alt="{\displaystyle k[x,y]\to k[t],\,f\mapsto f\left(t^{2},t^{3}\right).}"></dd></dl> In other words, it is the subalgebra of k[t] generated by t2 and t3. Example: let f be a polynomial in one variable, that is, an element in a polynomial ring R. Then f(x + h) is an element in R[h] and f(x + h) – f(x) is divisible by h in that ring. The result of substituting zero to h in (f(x + h) – f(x)) / h is f' (x), the derivative of f at x. The substitution is a special case of the universal property of a polynomial ring. The property states: given a ring homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :R\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mo>:</mo> <mi>R</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi :R\to S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/507c68fca8762e2a6e6aaa49dae7c0381fcc6525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.2ex; height:2.509ex;" alt="{\displaystyle \phi :R\to S}"> and an element x in S there exists a unique ring homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\phi }}:R[t]\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>:</mo> <mi>R</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\phi }}:R[t]\to S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9727858171f165d5705e60857e97068df5611bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.448ex; height:3.509ex;" alt="{\displaystyle {\overline {\phi }}:R[t]\to S}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\phi }}(t)=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\phi }}(t)=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ea7821d2f68704703f72ba7478aea23c483c47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.577ex; height:3.509ex;" alt="{\displaystyle {\overline {\phi }}(t)=x}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\phi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ϕ</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\phi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/19839359a0b8f888d007de4891a59f29984d41e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.5ex; height:3.343ex;" alt="{\displaystyle {\overline {\phi }}}"> restricts to ϕ.<a href="#cite_note-FOOTNOTEJacobson2009122Theorem_2.10-42">[37]</a> For example, choosing a basis, a <a href="/wiki/Symmetric_algebra" title="Symmetric algebra">symmetric algebra</a> satisfies the universal property and so is a polynomial ring. To give an example, let S be the ring of all functions from R to itself; the addition and the multiplication are those of functions. Let x be the identity function. Each r in R defines a constant function, giving rise to the homomorphism R → S. The universal property says that this map extends uniquely to <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[t]\to S,\quad f\mapsto {\overline {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mo stretchy="false">→</mo> <mi>S</mi> <mo>,</mo> <mspace width="1em" /> <mi>f</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[t]\to S,\quad f\mapsto {\overline {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15eb76de81b0c53d2b82343a721fc4d0cc6f6e29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.744ex; height:3.509ex;" alt="{\displaystyle R[t]\to S,\quad f\mapsto {\overline {f}}}"></dd></dl> (t maps to x) where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {f}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de4554a5933ca51745edacd34a5a6aaea9a9f4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.484ex; height:3.343ex;" alt="{\displaystyle {\overline {f}}}"> is the <a href="/wiki/Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial function</a> defined by f. The resulting map is injective if and only if R is infinite. Given a non-constant monic polynomial f in R[t], there exists a ring S containing R such that f is a product of linear factors in S[t].<a href="#cite_note-FOOTNOTEBourbaki1964Ch_5._§1,_Lemma_2-43">[38]</a> Let k be an algebraically closed field. The <a href="/wiki/Hilbert%27s_Nullstellensatz" title="Hilbert's Nullstellensatz">Hilbert's Nullstellensatz</a> (theorem of zeros) states that there is a natural one-to-one correspondence between the set of all prime ideals in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\left[t_{1},\ldots ,t_{n}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mrow> <mo>[</mo> <mrow> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\left[t_{1},\ldots ,t_{n}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/627d900f3df1fb1ea626778cc817791536cad857" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.022ex; height:2.843ex;" alt="{\displaystyle k\left[t_{1},\ldots ,t_{n}\right]}"> and the set of closed subvarieties of kn. In particular, many local problems in algebraic geometry may be attacked through the study of the generators of an ideal in a polynomial ring. (cf. <a href="/wiki/Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner basis</a>.) There are some other related constructions. A <a href="/wiki/Formal_power_series_ring" class="mw-redirect" title="Formal power series ring">formal power series ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[\![t]\!]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mspace width="negativethinmathspace" /> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mspace width="negativethinmathspace" /> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[\![t]\!]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b0ed61bc4efa03186f6fadccd730eb17bc9e0c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.417ex; height:2.843ex;" alt="{\displaystyle R[\![t]\!]}"> consists of formal power series <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{0}^{\infty }a_{i}t^{i},\quad a_{i}\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{0}^{\infty }a_{i}t^{i},\quad a_{i}\in R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a57a8f3478e3f57a12687fc46e781386df01791" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.402ex; height:6.843ex;" alt="{\displaystyle \sum _{0}^{\infty }a_{i}t^{i},\quad a_{i}\in R}"></dd></dl> together with multiplication and addition that mimic those for convergent series. It contains R[t] as a subring. A formal power series ring does not have the universal property of a polynomial ring; a series may not converge after a substitution. The important advantage of a formal power series ring over a polynomial ring is that it is <a href="/wiki/Local_ring" title="Local ring">local</a> (in fact, <a href="/wiki/Complete_ring" class="mw-redirect" title="Complete ring">complete</a>). <div class="mw-heading mw-heading3"><h3 id="Matrix_ring_and_endomorphism_ring">Matrix ring and endomorphism ring</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=27" title="Edit section: Matrix ring and endomorphism ring">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Matrix_ring" title="Matrix ring">Matrix ring</a> and <a href="/wiki/Endomorphism_ring" title="Endomorphism ring">Endomorphism ring</a></div> Let R be a ring (not necessarily commutative). The set of all square matrices of size n with entries in R forms a ring with the entry-wise addition and the usual matrix multiplication. It is called the <a href="/wiki/Matrix_ring" title="Matrix ring">matrix ring</a> and is denoted by Mn(R). Given a right R-module U, the set of all R-linear maps from U to itself forms a ring with addition that is of function and multiplication that is of <a href="/wiki/Composition_of_functions" class="mw-redirect" title="Composition of functions">composition of functions</a>; it is called the endomorphism ring of U and is denoted by EndR(U). As in linear algebra, a matrix ring may be canonically interpreted as an endomorphism ring: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {End} _{R}(R^{n})\simeq \operatorname {M} _{n}(R).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>End</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>≃</mo> <msub> <mi mathvariant="normal">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {End} _{R}(R^{n})\simeq \operatorname {M} _{n}(R).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1b83bafcf436ca385bcdb924441c123f20417fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.107ex; height:2.843ex;" alt="{\displaystyle \operatorname {End} _{R}(R^{n})\simeq \operatorname {M} _{n}(R).}"> This is a special case of the following fact: If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\oplus _{1}^{n}U\to \oplus _{1}^{n}U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <msubsup> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mi>U</mi> <mo stretchy="false">→</mo> <msubsup> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:\oplus _{1}^{n}U\to \oplus _{1}^{n}U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0794e4d294e2a0cb15b78e485cf39e825fd5a651" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.448ex; height:2.843ex;" alt="{\displaystyle f:\oplus _{1}^{n}U\to \oplus _{1}^{n}U}"> is an R-linear map, then f may be written as a matrix with entries fij in S = EndR(U), resulting in the ring isomorphism: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {End} _{R}(\oplus _{1}^{n}U)\to \operatorname {M} _{n}(S),\quad f\mapsto (f_{ij}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>End</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msubsup> <mo>⊕</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msubsup> <mi>U</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msub> <mi mathvariant="normal">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>f</mi> <mo stretchy="false">↦</mo> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {End} _{R}(\oplus _{1}^{n}U)\to \operatorname {M} _{n}(S),\quad f\mapsto (f_{ij}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/739fd79fe83d9e73b08e890d9e4558c40ceddff3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.86ex; height:3.009ex;" alt="{\displaystyle \operatorname {End} _{R}(\oplus _{1}^{n}U)\to \operatorname {M} _{n}(S),\quad f\mapsto (f_{ij}).}"></dd></dl> Any ring homomorphism R → S induces Mn(R) → Mn(S).<a href="#cite_note-FOOTNOTECohn20034.4-44">[39]</a> <a href="/wiki/Schur%27s_lemma" title="Schur's lemma">Schur's lemma</a> says that if U is a simple right R-module, then EndR(U) is a division ring.<a href="#cite_note-FOOTNOTELang2002Ch._XVII._Proposition_1.1-45">[40]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=\bigoplus _{i=1}^{r}U_{i}^{\oplus m_{i}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <munderover> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </munderover> <msubsup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊕</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\bigoplus _{i=1}^{r}U_{i}^{\oplus m_{i}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2dabc8e8c836b2e2aeffbbaf11292e519864d894" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.198ex; height:6.843ex;" alt="{\displaystyle U=\bigoplus _{i=1}^{r}U_{i}^{\oplus m_{i}}}"> is a direct sum of mi-copies of simple R-modules <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U_{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U_{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22cd8453a56ce4cfc76c1a23c80c47a6bbe00441" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.034ex; height:2.509ex;" alt="{\displaystyle U_{i},}"> then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {End} _{R}(U)\simeq \prod _{i=1}^{r}\operatorname {M} _{m_{i}}(\operatorname {End} _{R}(U_{i})).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>End</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>≃</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </munderover> <msub> <mi mathvariant="normal">M</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>End</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {End} _{R}(U)\simeq \prod _{i=1}^{r}\operatorname {M} _{m_{i}}(\operatorname {End} _{R}(U_{i})).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f63fe0fcbdba5aeeac0347644927bf7e58b99488" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:32.425ex; height:6.843ex;" alt="{\displaystyle \operatorname {End} _{R}(U)\simeq \prod _{i=1}^{r}\operatorname {M} _{m_{i}}(\operatorname {End} _{R}(U_{i})).}"></dd></dl> The <a href="/wiki/Artin%E2%80%93Wedderburn_theorem" class="mw-redirect" title="Artin–Wedderburn theorem">Artin–Wedderburn theorem</a> states any <a href="/wiki/Semisimple_ring" class="mw-redirect" title="Semisimple ring">semisimple ring</a> (cf. below) is of this form. A ring R and the matrix ring Mn(R) over it are <a href="/wiki/Morita_equivalent" class="mw-redirect" title="Morita equivalent">Morita equivalent</a>: the <a href="/wiki/Category_(mathematics)" title="Category (mathematics)">category</a> of right modules of R is equivalent to the category of right modules over Mn(R).<a href="#cite_note-FOOTNOTECohn20034.4-44">[39]</a> In particular, two-sided ideals in R correspond in one-to-one to two-sided ideals in Mn(R). <div class="mw-heading mw-heading3"><h3 id="Limits_and_colimits_of_rings">Limits and colimits of rings</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=28" title="Edit section: Limits and colimits of rings">edit</a>]</div> Let Ri be a sequence of rings such that Ri is a subring of Ri + 1 for all i. Then the union (or <a href="/wiki/Filtered_colimit" class="mw-redirect" title="Filtered colimit">filtered colimit</a>) of Ri is the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varinjlim R_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP"> <munder> <mi>lim</mi> <mo>→</mo> </munder> </mrow> <mo>⁡</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varinjlim R_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeca73f901302826b173c0d18522d64c254c2219" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.637ex; margin-bottom: -0.367ex; width:6.31ex; height:3.843ex;" alt="{\displaystyle \varinjlim R_{i}}"> defined as follows: it is the disjoint union of all Ri's modulo the equivalence relation x ~ y if and only if x = y in Ri for sufficiently large i. Examples of colimits: <ul><li>A polynomial ring in infinitely many variables: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[t_{1},t_{2},\cdots ]=\varinjlim R[t_{1},t_{2},\cdots ,t_{m}].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-OP"> <munder> <mi>lim</mi> <mo>→</mo> </munder> </mrow> <mo>⁡</mo> <mi>R</mi> <mo stretchy="false">[</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[t_{1},t_{2},\cdots ]=\varinjlim R[t_{1},t_{2},\cdots ,t_{m}].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c562ae4e64063e5aedc587c8442c596aed80eb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.637ex; margin-bottom: -0.367ex; width:34.701ex; height:4.009ex;" alt="{\displaystyle R[t_{1},t_{2},\cdots ]=\varinjlim R[t_{1},t_{2},\cdots ,t_{m}].}"></li> <li>The <a href="/wiki/Algebraic_closure" title="Algebraic closure">algebraic closure</a> of <a href="/wiki/Finite_field" title="Finite field">finite fields</a> of the same characteristic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {\mathbf {F} }}_{p}=\varinjlim \mathbf {F} _{p^{m}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-OP"> <munder> <mi>lim</mi> <mo>→</mo> </munder> </mrow> <mo>⁡</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">F</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {\mathbf {F} }}_{p}=\varinjlim \mathbf {F} _{p^{m}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/976e381fefbbf86d0cf9b9d9ec400b1003c1b227" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.637ex; margin-bottom: -0.367ex; width:14.426ex; height:4.676ex;" alt="{\displaystyle {\overline {\mathbf {F} }}_{p}=\varinjlim \mathbf {F} _{p^{m}}.}"></li> <li>The field of <a href="/wiki/Formal_Laurent_series" class="mw-redirect" title="Formal Laurent series">formal Laurent series</a> over a field k: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(\!(t)\!)=\varinjlim t^{-m}k[\![t]\!]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">(</mo> <mspace width="negativethinmathspace" /> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="negativethinmathspace" /> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-OP"> <munder> <mi>lim</mi> <mo>→</mo> </munder> </mrow> <mo>⁡</mo> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>m</mi> </mrow> </msup> <mi>k</mi> <mo stretchy="false">[</mo> <mspace width="negativethinmathspace" /> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mspace width="negativethinmathspace" /> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k(\!(t)\!)=\varinjlim t^{-m}k[\![t]\!]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/536bed6a210e67bcc910b2055075b9afa8306105" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.637ex; margin-bottom: -0.367ex; width:19.398ex; height:4.176ex;" alt="{\displaystyle k(\!(t)\!)=\varinjlim t^{-m}k[\![t]\!]}"> (it is the field of fractions of the <a href="/wiki/Formal_power_series_ring" class="mw-redirect" title="Formal power series ring">formal power series ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[\![t]\!].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">[</mo> <mspace width="negativethinmathspace" /> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mspace width="negativethinmathspace" /> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k[\![t]\!].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db5e2887e456f95fb545b62435a54ac319e7a782" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.511ex; height:2.843ex;" alt="{\displaystyle k[\![t]\!].}">)</li> <li>The <a href="/wiki/Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function field of an algebraic variety</a> over a field k is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varinjlim k[U]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP"> <munder> <mi>lim</mi> <mo>→</mo> </munder> </mrow> <mo>⁡</mo> <mi>k</mi> <mo stretchy="false">[</mo> <mi>U</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varinjlim k[U]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc7735d64bbdf3d5a25544d60bea11fdd0a5f54e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.637ex; margin-bottom: -0.367ex; width:8.034ex; height:4.009ex;" alt="{\displaystyle \varinjlim k[U]}"> where the limit runs over all the coordinate rings k[U] of nonempty open subsets U (more succinctly it is the <a href="/wiki/Stalk_(mathematics)" class="mw-redirect" title="Stalk (mathematics)">stalk</a> of the structure sheaf at the <a href="/wiki/Generic_point" title="Generic point">generic point</a>.)</li></ul> Any commutative ring is the colimit of <a href="/wiki/Finitely_generated_ring" class="mw-redirect" title="Finitely generated ring">finitely generated subrings</a>. A <a href="/wiki/Projective_limit" class="mw-redirect" title="Projective limit">projective limit</a> (or a <a href="/wiki/Filtered_limit" class="mw-redirect" title="Filtered limit">filtered limit</a>) of rings is defined as follows. Suppose we are given a family of rings Ri, i running over positive integers, say, and ring homomorphisms Rj → Ri, j ≥ i such that Ri → Ri are all the identities and Rk → Rj → Ri is Rk → Ri whenever k ≥ j ≥ i. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varprojlim R_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-OP"> <munder> <mi>lim</mi> <mo>←</mo> </munder> </mrow> <mo>⁡</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \varprojlim R_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75f7b0aaf3560c48ed9d014f38367282ad2b06e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.637ex; margin-bottom: -0.367ex; width:6.375ex; height:3.843ex;" alt="{\displaystyle \varprojlim R_{i}}"> is the subring of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \prod R_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∏</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \prod R_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a48ed081fc258b24f86887e97d214d0a1c19ef7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.144ex; height:2.843ex;" alt="{\displaystyle \textstyle \prod R_{i}}"> consisting of (xn) such that xj maps to xi under Rj → Ri, j ≥ i. For an example of a projective limit, see <a href="#Completion">§ Completion</a>. <div class="mw-heading mw-heading3"><h3 id="Localization">Localization</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=29" title="Edit section: Localization">edit</a>]</div> The <a href="/wiki/Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">localization</a> generalizes the construction of the <a href="/wiki/Field_of_fractions" title="Field of fractions">field of fractions</a> of an integral domain to an arbitrary ring and modules. Given a (not necessarily commutative) ring R and a subset S of R, there exists a ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[S^{-1}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[S^{-1}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc8e9aca12e1c82bcb76786e721b503057d121bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.912ex; height:3.176ex;" alt="{\displaystyle R[S^{-1}]}"> together with the ring homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\to R\left[S^{-1}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">→</mo> <mi>R</mi> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\to R\left[S^{-1}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff24251e09611253a5e59710f41fef5e9ea8295" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.323ex; height:3.343ex;" alt="{\displaystyle R\to R\left[S^{-1}\right]}"> that "inverts" S; that is, the homomorphism maps elements in S to unit elements in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\left[S^{-1}\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\left[S^{-1}\right],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc5da3880f2a84e3c59d744d4a6a70c91648127f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.979ex; height:3.343ex;" alt="{\displaystyle R\left[S^{-1}\right],}"> and, moreover, any ring homomorphism from R that "inverts" S uniquely factors through <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\left[S^{-1}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\left[S^{-1}\right].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/477d928832e71936d39c6e196a1ec4c7f9c5f003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.979ex; height:3.343ex;" alt="{\displaystyle R\left[S^{-1}\right].}"><a href="#cite_note-FOOTNOTECohn1995Proposition_1.3.1-46">[41]</a> The ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\left[S^{-1}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\left[S^{-1}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2fa5e7650f1b84116f5dd0986162ed090cafea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.945ex; height:3.343ex;" alt="{\displaystyle R\left[S^{-1}\right]}"> is called the localization of R with respect to S. For example, if R is a commutative ring and f an element in R, then the localization <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\left[f^{-1}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow> <mo>[</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\left[f^{-1}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/866a1856d4f363ae6d4d2f3818ff980b72442c51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.744ex; height:3.343ex;" alt="{\displaystyle R\left[f^{-1}\right]}"> consists of elements of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r/f^{n},\,r\in R,\,n\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>,</mo> <mspace width="thinmathspace" /> <mi>r</mi> <mo>∈</mo> <mi>R</mi> <mo>,</mo> <mspace width="thinmathspace" /> <mi>n</mi> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r/f^{n},\,r\in R,\,n\geq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a08b51517face7da73d55a6c17a9e6197cd06bca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.901ex; height:2.843ex;" alt="{\displaystyle r/f^{n},\,r\in R,\,n\geq 0}"> (to be precise, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\left[f^{-1}\right]=R[t]/(tf-1).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow> <mo>[</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <mi>R</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mi>f</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\left[f^{-1}\right]=R[t]/(tf-1).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/751001e38a83d034e0cf683b63cf45ac724151ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.479ex; height:3.343ex;" alt="{\displaystyle R\left[f^{-1}\right]=R[t]/(tf-1).}">)<a href="#cite_note-FOOTNOTEEisenbud1995Exercise_2.2-47">[42]</a> The localization is frequently applied to a commutative ring R with respect to the complement of a prime ideal (or a union of prime ideals) in R. In that case <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=R-{\mathfrak {p}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>R</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=R-{\mathfrak {p}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36bdc81564f7beca745923b53702413398b396b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.011ex; height:2.509ex;" alt="{\displaystyle S=R-{\mathfrak {p}},}"> one often writes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124aee5f6d80492d2746e652dee942c36ef6e1c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.818ex; height:2.843ex;" alt="{\displaystyle R_{\mathfrak {p}}}"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\left[S^{-1}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\left[S^{-1}\right].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/477d928832e71936d39c6e196a1ec4c7f9c5f003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.979ex; height:3.343ex;" alt="{\displaystyle R\left[S^{-1}\right].}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124aee5f6d80492d2746e652dee942c36ef6e1c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.818ex; height:2.843ex;" alt="{\displaystyle R_{\mathfrak {p}}}"> is then a <a href="/wiki/Local_ring" title="Local ring">local ring</a> with the <a href="/wiki/Maximal_ideal" title="Maximal ideal">maximal ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}R_{\mathfrak {p}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}R_{\mathfrak {p}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/582a26f263195247f30fa06dad60f8d88f26c164" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.628ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {p}}R_{\mathfrak {p}}.}"> This is the reason for the terminology "localization". The field of fractions of an integral domain R is the localization of R at the prime ideal zero. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}"> is a prime ideal of a commutative ring R, then the field of fractions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R/{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R/{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9da6e6f1e63ba7c169bec572ae2ca16f1c58936" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.089ex; height:2.843ex;" alt="{\displaystyle R/{\mathfrak {p}}}"> is the same as the residue field of the local ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/124aee5f6d80492d2746e652dee942c36ef6e1c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.818ex; height:2.843ex;" alt="{\displaystyle R_{\mathfrak {p}}}"> and is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k({\mathfrak {p}}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k({\mathfrak {p}}).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d44d0082dc4ec6a877af31dc1217fcab471f7e94" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.83ex; height:2.843ex;" alt="{\displaystyle k({\mathfrak {p}}).}"> If M is a left R-module, then the localization of M with respect to S is given by a <a href="/wiki/Change_of_rings" title="Change of rings">change of rings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\left[S^{-1}\right]=R\left[S^{-1}\right]\otimes _{R}M.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <mi>R</mi> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mi>M</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\left[S^{-1}\right]=R\left[S^{-1}\right]\otimes _{R}M.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/182e09cf0feec2967d8337e197459da4d0084bde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.075ex; height:3.343ex;" alt="{\displaystyle M\left[S^{-1}\right]=R\left[S^{-1}\right]\otimes _{R}M.}"> The most important properties of localization are the following: when R is a commutative ring and S a multiplicatively closed subset <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}\mapsto {\mathfrak {p}}\left[S^{-1}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}\mapsto {\mathfrak {p}}\left[S^{-1}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5dc99eed581db91f5f4d659076ee275ba9728331" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.12ex; height:3.343ex;" alt="{\displaystyle {\mathfrak {p}}\mapsto {\mathfrak {p}}\left[S^{-1}\right]}"> is a bijection between the set of all prime ideals in R disjoint from S and the set of all prime ideals in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\left[S^{-1}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\left[S^{-1}\right].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/477d928832e71936d39c6e196a1ec4c7f9c5f003" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.979ex; height:3.343ex;" alt="{\displaystyle R\left[S^{-1}\right].}"><a href="#cite_note-FOOTNOTEMilne2012Proposition_6.4-48">[43]</a></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\left[S^{-1}\right]=\varinjlim R\left[f^{-1}\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-OP"> <munder> <mi>lim</mi> <mo>→</mo> </munder> </mrow> <mo>⁡</mo> <mi>R</mi> <mrow> <mo>[</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\left[S^{-1}\right]=\varinjlim R\left[f^{-1}\right],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a32ebca63604a2aebf74c48d3b5f3935c419df84" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.637ex; margin-bottom: -0.367ex; width:23.568ex; height:4.343ex;" alt="{\displaystyle R\left[S^{-1}\right]=\varinjlim R\left[f^{-1}\right],}"> f running over elements in S with partial ordering given by divisibility.<a href="#cite_note-FOOTNOTEMilne2012end_of_Chapter_7-49">[44]</a></li> <li>The localization is exact: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to M'\left[S^{-1}\right]\to M\left[S^{-1}\right]\to M''\left[S^{-1}\right]\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <msup> <mi>M</mi> <mo>′</mo> </msup> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo stretchy="false">→</mo> <mi>M</mi> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo stretchy="false">→</mo> <msup> <mi>M</mi> <mo>″</mo> </msup> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to M'\left[S^{-1}\right]\to M\left[S^{-1}\right]\to M''\left[S^{-1}\right]\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/645fb810cb3b83f73d5257efcb9871cde5ddd25a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:44.585ex; height:3.343ex;" alt="{\displaystyle 0\to M'\left[S^{-1}\right]\to M\left[S^{-1}\right]\to M''\left[S^{-1}\right]\to 0}"> is exact over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\left[S^{-1}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\left[S^{-1}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2fa5e7650f1b84116f5dd0986162ed090cafea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.945ex; height:3.343ex;" alt="{\displaystyle R\left[S^{-1}\right]}"> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to M'\to M\to M''\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <msup> <mi>M</mi> <mo>′</mo> </msup> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">→</mo> <msup> <mi>M</mi> <mo>″</mo> </msup> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to M'\to M\to M''\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a12da2cdc62593c2b3d9e1a796698a052f8ccab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:26.043ex; height:2.509ex;" alt="{\displaystyle 0\to M'\to M\to M''\to 0}"> is exact over R.</li> <li>Conversely, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to M'_{\mathfrak {m}}\to M_{\mathfrak {m}}\to M''_{\mathfrak {m}}\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> <mo>′</mo> </msubsup> <mo stretchy="false">→</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </msub> <mo stretchy="false">→</mo> <msubsup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> <mo>″</mo> </msubsup> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to M'_{\mathfrak {m}}\to M_{\mathfrak {m}}\to M''_{\mathfrak {m}}\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6599b67cf1f3b3fd12126b3e010da58fa550b58e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.022ex; height:2.509ex;" alt="{\displaystyle 0\to M'_{\mathfrak {m}}\to M_{\mathfrak {m}}\to M''_{\mathfrak {m}}\to 0}"> is exact for any maximal ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95c2e12edac0425066c23a2fb3f6dc66c9498aea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.429ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {m}},}"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\to M'\to M\to M''\to 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo stretchy="false">→</mo> <msup> <mi>M</mi> <mo>′</mo> </msup> <mo stretchy="false">→</mo> <mi>M</mi> <mo stretchy="false">→</mo> <msup> <mi>M</mi> <mo>″</mo> </msup> <mo stretchy="false">→</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\to M'\to M\to M''\to 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a12da2cdc62593c2b3d9e1a796698a052f8ccab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:26.043ex; height:2.509ex;" alt="{\displaystyle 0\to M'\to M\to M''\to 0}"> is exact.</li> <li>A remark: localization is no help in proving a global existence. One instance of this is that if two modules are isomorphic at all prime ideals, it does not follow that they are isomorphic. (One way to explain this is that the localization allows one to view a module as a sheaf over prime ideals and a sheaf is inherently a local notion.)</li></ul> In <a href="/wiki/Category_theory" title="Category theory">category theory</a>, a <a href="/wiki/Localization_of_a_category" title="Localization of a category">localization of a category</a> amounts to making some morphisms isomorphisms. An element in a commutative ring R may be thought of as an endomorphism of any R-module. Thus, categorically, a localization of R with respect to a subset S of R is a <a href="/wiki/Functor" title="Functor">functor</a> from the category of R-modules to itself that sends elements of S viewed as endomorphisms to automorphisms and is universal with respect to this property. (Of course, R then maps to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\left[S^{-1}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\left[S^{-1}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2fa5e7650f1b84116f5dd0986162ed090cafea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.945ex; height:3.343ex;" alt="{\displaystyle R\left[S^{-1}\right]}"> and R-modules map to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\left[S^{-1}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow> <mo>[</mo> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\left[S^{-1}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2fa5e7650f1b84116f5dd0986162ed090cafea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.945ex; height:3.343ex;" alt="{\displaystyle R\left[S^{-1}\right]}">-modules.) <div class="mw-heading mw-heading3"><h3 id="Completion">Completion</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=30" title="Edit section: Completion">edit</a>]</div> Let R be a commutative ring, and let I be an ideal of R. The <a href="/wiki/Completion_(ring_theory)" class="mw-redirect" title="Completion (ring theory)">completion</a> of R at I is the projective limit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {R}}=\varprojlim R/I^{n};}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-OP"> <munder> <mi>lim</mi> <mo>←</mo> </munder> </mrow> <mo>⁡</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {R}}=\varprojlim R/I^{n};}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e1290ad833e81e41c7b9a8d12646957943e826e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.637ex; margin-bottom: -0.367ex; width:14.682ex; height:4.509ex;" alt="{\displaystyle {\hat {R}}=\varprojlim R/I^{n};}"> it is a commutative ring. The canonical homomorphisms from R to the quotients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R/I^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R/I^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/450dc301658614a35b0cc329114812e4055cf50f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.361ex; height:2.843ex;" alt="{\displaystyle R/I^{n}}"> induce a homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\to {\hat {R}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\to {\hat {R}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bcf6bbeccf935f3bf76ec426682a07f51a454da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.789ex; height:2.843ex;" alt="{\displaystyle R\to {\hat {R}}.}"> The latter homomorphism is injective if R is a Noetherian integral domain and I is a proper ideal, or if R is a Noetherian local ring with maximal ideal I, by <a href="/wiki/Krull%27s_intersection_theorem" class="mw-redirect" title="Krull's intersection theorem">Krull's intersection theorem</a>.<a href="#cite_note-FOOTNOTEAtiyahMacdonald1969Theorem_10.17_and_its_corollaries-50">[45]</a> The construction is especially useful when I is a maximal ideal. The basic example is the completion of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">⁠ at the principal ideal (p) generated by a prime number p; it is called the ring of <a href="/wiki/P-adic_integer" class="mw-redirect" title="P-adic integer">p-adic integers</a> and is denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d1fd592239d60f9a9f36feb4b3be40df666f6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.256ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}.}">⁠ The completion can in this case be constructed also from the <a href="/wiki/P-adic_absolute_value" class="mw-redirect" title="P-adic absolute value">p-adic absolute value</a> on ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869719f08f506bf866043442858fb3da1d4b4b5b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.455ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} .}">⁠ The p-adic absolute value on ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">⁠ is a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\mapsto |x|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\mapsto |x|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04f7cb7bcc9c126b2d18614065cfeeb20f8e5514" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.567ex; height:2.843ex;" alt="{\displaystyle x\mapsto |x|}"> from ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">⁠ to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">⁠ given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |n|_{p}=p^{-v_{p}(n)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |n|_{p}=p^{-v_{p}(n)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/675d39c82a1507cf2905e97039be955bdc0b203b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.425ex; height:3.676ex;" alt="{\displaystyle |n|_{p}=p^{-v_{p}(n)}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{p}(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{p}(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab7ef07cef84d44d3fd868891d304b7afafae1b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.391ex; height:3.009ex;" alt="{\displaystyle v_{p}(n)}"> denotes the exponent of p in the prime factorization of a nonzero integer n into prime numbers (we also put <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |0|_{p}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mn>0</mn> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |0|_{p}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f790cbc348f0929770ccd86dac7e18f5b2644930" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.776ex; height:3.176ex;" alt="{\displaystyle |0|_{p}=0}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |m/n|_{p}=|m|_{p}/|n|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>m</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>n</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |m/n|_{p}=|m|_{p}/|n|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/790be7ff106c016200e34693cce080e32924ab1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.352ex; height:3.176ex;" alt="{\displaystyle |m/n|_{p}=|m|_{p}/|n|_{p}}">). It defines a distance function on ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">⁠ and the completion of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5909f0b54e4718fa24d5fd34d54189d24a66e9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }">⁠ as a <a href="/wiki/Metric_space" title="Metric space">metric space</a> is denoted by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb4b0959c6687de4ffc0158e0b1d35d2137774d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.514ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}.}">⁠ It is again a field since the field operations extend to the completion. The subring of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}">⁠ consisting of elements x with |x|p ≤ 1 is isomorphic to ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2d1fd592239d60f9a9f36feb4b3be40df666f6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.256ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}.}">⁠ Similarly, the formal power series ring R[{[t]}] is the completion of R[t] at (t) (see also <a href="/wiki/Hensel%27s_lemma" title="Hensel's lemma">Hensel's lemma</a>) A complete ring has much simpler structure than a commutative ring. This owns to the <a href="/wiki/Cohen_structure_theorem" title="Cohen structure theorem">Cohen structure theorem</a>, which says, roughly, that a complete local ring tends to look like a formal power series ring or a quotient of it. On the other hand, the interaction between the <a href="/wiki/Integral_closure" class="mw-redirect" title="Integral closure">integral closure</a> and completion has been among the most important aspects that distinguish modern commutative ring theory from the classical one developed by the likes of Noether. Pathological examples found by Nagata led to the reexamination of the roles of Noetherian rings and motivated, among other things, the definition of <a href="/wiki/Excellent_ring" title="Excellent ring">excellent ring</a>. <div class="mw-heading mw-heading3"><h3 id="Rings_with_generators_and_relations">Rings with generators and relations</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=31" title="Edit section: Rings with generators and relations">edit</a>]</div> The most general way to construct a ring is by specifying generators and relations. Let F be a <a href="/wiki/Free_ring" class="mw-redirect" title="Free ring">free ring</a> (that is, free algebra over the integers) with the set X of symbols, that is, F consists of polynomials with integral coefficients in noncommuting variables that are elements of X. A free ring satisfies the universal property: any function from the set X to a ring R factors through F so that F → R is the unique ring homomorphism. Just as in the group case, every ring can be represented as a quotient of a free ring.<a href="#cite_note-FOOTNOTECohn1995[httpsbooksgooglecombooksidu-4ADgUgpSMCpgPA242_pg._242]-51">[46]</a> Now, we can impose relations among symbols in X by taking a quotient. Explicitly, if E is a subset of F, then the quotient ring of F by the ideal generated by E is called the ring with generators X and relations E. If we used a ring, say, A as a base ring instead of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3aa4cb112cbe4f94a3ff8569f869c31dce5fce4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.197ex; height:2.509ex;" alt="{\displaystyle \mathbb {Z} ,}">⁠ then the resulting ring will be over A. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\{xy-yx\mid x,y\in X\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mi>y</mi> <mo>−</mo> <mi>y</mi> <mi>x</mi> <mo>∣</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>X</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\{xy-yx\mid x,y\in X\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b34960c4750ac9c3844ff164c38a233d41d92367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.934ex; height:2.843ex;" alt="{\displaystyle E=\{xy-yx\mid x,y\in X\},}"> then the resulting ring will be the usual polynomial ring with coefficients in A in variables that are elements of X (It is also the same thing as the <a href="/wiki/Symmetric_algebra" title="Symmetric algebra">symmetric algebra</a> over A with symbols X.) In the category-theoretic terms, the formation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\mapsto {\text{the free ring generated by the set }}S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">↦</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>the free ring generated by the set </mtext> </mrow> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\mapsto {\text{the free ring generated by the set }}S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1bd4c70bcd65ed54c5ce2b3d34ffdcc3895e89ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:40.035ex; height:2.509ex;" alt="{\displaystyle S\mapsto {\text{the free ring generated by the set }}S}"> is the left adjoint functor of the <a href="/wiki/Forgetful_functor" title="Forgetful functor">forgetful functor</a> from the <a href="/wiki/Category_of_rings" title="Category of rings">category of rings</a> to Set (and it is often called the free ring functor.) Let A, B be algebras over a commutative ring R. Then the tensor product of R-modules <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\otimes _{R}B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\otimes _{R}B}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4773c5f4d6541ec487737f1c3962e1aafb7fe55" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.827ex; height:2.509ex;" alt="{\displaystyle A\otimes _{R}B}"> is an R-algebra with multiplication characterized by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x\otimes u)(y\otimes v)=xy\otimes uv.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⊗</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>⊗</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mi>y</mi> <mo>⊗</mo> <mi>u</mi> <mi>v</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x\otimes u)(y\otimes v)=xy\otimes uv.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae8c6a4f66f11f6c77cb20c78d91b6c5a26b7f10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.77ex; height:2.843ex;" alt="{\displaystyle (x\otimes u)(y\otimes v)=xy\otimes uv.}"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Tensor_product_of_algebras" title="Tensor product of algebras">Tensor product of algebras</a> and <a href="/wiki/Change_of_rings" title="Change of rings">Change of rings</a></div> <div class="mw-heading mw-heading2"><h2 id="Special_kinds_of_rings">Special kinds of rings</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=32" title="Edit section: Special kinds of rings">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Domains">Domains</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=33" title="Edit section: Domains">edit</a>]</div> A <a href="/wiki/Zero_ring" title="Zero ring">nonzero</a> ring with no nonzero <a href="/wiki/Zero-divisor" class="mw-redirect" title="Zero-divisor">zero-divisors</a> is called a <a href="/wiki/Domain_(ring_theory)" title="Domain (ring theory)">domain</a>. A commutative domain is called an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>. The most important integral domains are principal ideal domains, PIDs for short, and fields. A principal ideal domain is an integral domain in which every ideal is principal. An important class of integral domains that contain a PID is a <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization domain</a> (UFD), an integral domain in which every nonunit element is a product of <a href="/wiki/Prime_element" title="Prime element">prime elements</a> (an element is prime if it generates a <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a>.) The fundamental question in <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a> is on the extent to which the <a href="/wiki/Ring_of_integers" title="Ring of integers">ring of (generalized) integers</a> in a <a href="/wiki/Number_field" class="mw-redirect" title="Number field">number field</a>, where an "ideal" admits prime factorization, fails to be a PID. Among theorems concerning a PID, the most important one is the <a href="/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain" title="Structure theorem for finitely generated modules over a principal ideal domain">structure theorem for finitely generated modules over a principal ideal domain</a>. The theorem may be illustrated by the following application to linear algebra.<a href="#cite_note-FOOTNOTELang2002Ch_XIV,_§2-52">[47]</a> Let V be a finite-dimensional vector space over a field k and f : V → V a linear map with minimal polynomial q. Then, since k[t] is a unique factorization domain, q factors into powers of distinct irreducible polynomials (that is, prime elements): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q=p_{1}^{e_{1}}\ldots p_{s}^{e_{s}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msubsup> <mo>…</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=p_{1}^{e_{1}}\ldots p_{s}^{e_{s}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f28851de1068565f9fc8372bc4d86b8766e31ea7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.27ex; height:3.176ex;" alt="{\displaystyle q=p_{1}^{e_{1}}\ldots p_{s}^{e_{s}}.}"> Letting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t\cdot v=f(v),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>⋅</mo> <mi>v</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>v</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\cdot v=f(v),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5206ffecae8cd986993c59935a9a90a256f37b71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.607ex; height:2.843ex;" alt="{\displaystyle t\cdot v=f(v),}"> we make V a k[t]-module. The structure theorem then says V is a direct sum of <a href="/wiki/Cyclic_module" title="Cyclic module">cyclic modules</a>, each of which is isomorphic to the module of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[t]/\left(p_{i}^{k_{j}}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow> <mo>(</mo> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msubsup> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k[t]/\left(p_{i}^{k_{j}}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20d1f6688eda0ad0b9ab06a59cc999e376b2b3fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.676ex; height:4.843ex;" alt="{\displaystyle k[t]/\left(p_{i}^{k_{j}}\right).}"> Now, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}(t)=t-\lambda _{i},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{i}(t)=t-\lambda _{i},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb23afca7f93a8402603789615154ac7e849252e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:14.288ex; height:2.843ex;" alt="{\displaystyle p_{i}(t)=t-\lambda _{i},}"> then such a cyclic module (for pi) has a basis in which the restriction of f is represented by a <a href="/wiki/Jordan_matrix" title="Jordan matrix">Jordan matrix</a>. Thus, if, say, k is algebraically closed, then all pi's are of the form t – λi and the above decomposition corresponds to the <a href="/wiki/Jordan_canonical_form" class="mw-redirect" title="Jordan canonical form">Jordan canonical form</a> of f. <figure typeof="mw:File/Thumb"><a href="/wiki/File:Ringhierarchy.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Ringhierarchy.png/272px-Ringhierarchy.png" decoding="async" width="272" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Ringhierarchy.png/408px-Ringhierarchy.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Ringhierarchy.png/544px-Ringhierarchy.png 2x" data-file-width="1280" data-file-height="720" /></a><figcaption>Hierarchy of several classes of rings with examples.</figcaption></figure> In algebraic geometry, UFDs arise because of smoothness. More precisely, a point in a variety (over a perfect field) is smooth if the local ring at the point is a <a href="/wiki/Regular_local_ring" title="Regular local ring">regular local ring</a>. A regular local ring is a UFD.<a href="#cite_note-FOOTNOTEWeibel2013[httpsbooksgooglecombooksidJa8xAAAAQBAJpgPA26_26]Ch_1,_Theorem_3.8-53">[48]</a> The following is a chain of <a href="/wiki/Subclass_(set_theory)" title="Subclass (set theory)">class inclusions</a> that describes the relationship between rings, domains and fields: <dl><dd><a href="/wiki/Rng_(algebra)" title="Rng (algebra)">rngs</a> ⊃ <a class="mw-selflink selflink">rings</a> ⊃ <a href="/wiki/Commutative_ring" title="Commutative ring">commutative rings</a> ⊃ <a href="/wiki/Integral_domain" title="Integral domain">integral domains</a> ⊃ <a href="/wiki/Integrally_closed_domain" title="Integrally closed domain">integrally closed domains</a> ⊃ <a href="/wiki/GCD_domain" title="GCD domain">GCD domains</a> ⊃ <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">unique factorization domains</a> ⊃ <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domains</a> ⊃ <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domains</a> ⊃ <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> ⊃ <a href="/wiki/Algebraically_closed_field" title="Algebraically closed field">algebraically closed fields</a></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Division_ring">Division ring</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=34" title="Edit section: Division ring">edit</a>]</div> A <a href="/wiki/Division_ring" title="Division ring">division ring</a> is a ring such that every non-zero element is a unit. A commutative division ring is a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>. A prominent example of a division ring that is not a field is the ring of <a href="/wiki/Quaternion" title="Quaternion">quaternions</a>. Any centralizer in a division ring is also a division ring. In particular, the center of a division ring is a field. It turned out that every finite domain (in particular finite division ring) is a field; in particular commutative (the <a href="/wiki/Wedderburn%27s_little_theorem" title="Wedderburn's little theorem">Wedderburn's little theorem</a>). Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field. The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the <a href="/wiki/Cartan%E2%80%93Brauer%E2%80%93Hua_theorem" title="Cartan–Brauer–Hua theorem">Cartan–Brauer–Hua theorem</a>. A <a href="/wiki/Cyclic_algebra" title="Cyclic algebra">cyclic algebra</a>, introduced by <a href="/wiki/L._E._Dickson" class="mw-redirect" title="L. E. Dickson">L. E. Dickson</a>, is a generalization of a <a href="/wiki/Quaternion_algebra" title="Quaternion algebra">quaternion algebra</a>. <div class="mw-heading mw-heading3"><h3 id="Semisimple_rings">Semisimple rings</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=35" title="Edit section: Semisimple rings">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Semisimple_module" title="Semisimple module">Semisimple module</a></div> A <a href="/wiki/Semisimple_module" title="Semisimple module">semisimple module</a> is a direct sum of simple modules. A <a href="/wiki/Semisimple_ring" class="mw-redirect" title="Semisimple ring">semisimple ring</a> is a ring that is semisimple as a left module (or right module) over itself. <div class="mw-heading mw-heading4"><h4 id="Examples">Examples</h4>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=36" title="Edit section: Examples">edit</a>]</div> <ul><li>A <a href="/wiki/Division_ring" title="Division ring">division ring</a> is semisimple (and <a href="/wiki/Simple_ring" title="Simple ring">simple</a>).</li> <li>For any division ring D and positive integer n, the matrix ring Mn(D) is semisimple (and <a href="/wiki/Simple_ring" title="Simple ring">simple</a>).</li> <li>For a field k and finite group G, the group ring kG is semisimple if and only if the <a href="/wiki/Characteristic_(algebra)" title="Characteristic (algebra)">characteristic</a> of k does not divide the <a href="/wiki/Order_(algebra)" class="mw-redirect" title="Order (algebra)">order</a> of G (<a href="/wiki/Maschke%27s_theorem" title="Maschke's theorem">Maschke's theorem</a>).</li> <li><a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebras</a> are semisimple.</li></ul> The <a href="/wiki/Weyl_algebra" title="Weyl algebra">Weyl algebra</a> over a field is a <a href="/wiki/Simple_ring" title="Simple ring">simple ring</a>, but it is not semisimple. The same holds for a <a href="/wiki/Differential_operator#Ring_of_multivariate_polynomial_differential_operators" title="Differential operator">ring of differential operators in many variables</a>. <div class="mw-heading mw-heading4"><h4 id="Properties">Properties</h4>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=37" title="Edit section: Properties">edit</a>]</div> Any module over a semisimple ring is semisimple. (Proof: A free module over a semisimple ring is semisimple and any module is a quotient of a free module.) For a ring R, the following are equivalent: <ul><li>R is semisimple.</li> <li>R is <a href="/wiki/Artinian_ring" title="Artinian ring">artinian</a> and <a href="/wiki/Semiprimitive_ring" title="Semiprimitive ring">semiprimitive</a>.</li> <li>R is a finite <a href="/wiki/Direct_product" title="Direct product">direct product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \prod _{i=1}^{r}\operatorname {M} _{n_{i}}(D_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </munderover> <msub> <mi mathvariant="normal">M</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \prod _{i=1}^{r}\operatorname {M} _{n_{i}}(D_{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b282119601ed34d92b79364c1d79e70a21d68ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.989ex; height:3.176ex;" alt="{\textstyle \prod _{i=1}^{r}\operatorname {M} _{n_{i}}(D_{i})}"> where each ni is a positive integer, and each Di is a division ring (<a href="/wiki/Artin%E2%80%93Wedderburn_theorem" class="mw-redirect" title="Artin–Wedderburn theorem">Artin–Wedderburn theorem</a>).</li></ul> Semisimplicity is closely related to separability. A unital associative algebra A over a field k is said to be <a href="/wiki/Separable_algebra" title="Separable algebra">separable</a> if the base extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\otimes _{k}F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\otimes _{k}F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b94ca94291b3216fd6f42f2624996c75350e583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.413ex; height:2.509ex;" alt="{\displaystyle A\otimes _{k}F}"> is semisimple for every <a href="/wiki/Field_extension" title="Field extension">field extension</a> F / k. If A happens to be a field, then this is equivalent to the usual definition in field theory (cf. <a href="/wiki/Separable_extension" title="Separable extension">separable extension</a>.) <div class="mw-heading mw-heading3"><h3 id="Central_simple_algebra_and_Brauer_group">Central simple algebra and Brauer group</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=38" title="Edit section: Central simple algebra and Brauer group">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Central_simple_algebra" title="Central simple algebra">Central simple algebra</a></div> For a field k, a k-algebra is central if its center is k and is simple if it is a <a href="/wiki/Simple_ring" title="Simple ring">simple ring</a>. Since the center of a simple k-algebra is a field, any simple k-algebra is a central simple algebra over its center. In this section, a central simple algebra is assumed to have finite dimension. Also, we mostly fix the base field; thus, an algebra refers to a k-algebra. The matrix ring of size n over a ring R will be denoted by Rn. The <a href="/wiki/Skolem%E2%80%93Noether_theorem" title="Skolem–Noether theorem">Skolem–Noether theorem</a> states any automorphism of a central simple algebra is inner. Two central simple algebras A and B are said to be similar if there are integers n and m such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\otimes _{k}k_{n}\approx B\otimes _{k}k_{m}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>≈</mo> <mi>B</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\otimes _{k}k_{n}\approx B\otimes _{k}k_{m}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8598a9947bfac9e5a3e9fcdc99e4a4882f427a03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.427ex; height:2.509ex;" alt="{\displaystyle A\otimes _{k}k_{n}\approx B\otimes _{k}k_{m}.}"><a href="#cite_note-FOOTNOTEMilneCFTCh_IV,_§2-54">[49]</a> Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{n}\otimes _{k}k_{m}\simeq k_{nm},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>≃</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>m</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{n}\otimes _{k}k_{m}\simeq k_{nm},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6de24e63d8e35c23a05a860a78f57c58fac314ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.863ex; height:2.509ex;" alt="{\displaystyle k_{n}\otimes _{k}k_{m}\simeq k_{nm},}"> the similarity is an equivalence relation. The similarity classes [A] with the multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [A][B]=\left[A\otimes _{k}B\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>A</mi> <mo stretchy="false">]</mo> <mo stretchy="false">[</mo> <mi>B</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mi>A</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>B</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [A][B]=\left[A\otimes _{k}B\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9181c4c9330a227ae8d9ffaabb33c18ed73459d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.923ex; height:2.843ex;" alt="{\displaystyle [A][B]=\left[A\otimes _{k}B\right]}"> form an abelian group called the <a href="/wiki/Brauer_group" title="Brauer group">Brauer group</a> of k and is denoted by Br(k). By the <a href="/wiki/Artin%E2%80%93Wedderburn_theorem" class="mw-redirect" title="Artin–Wedderburn theorem">Artin–Wedderburn theorem</a>, a central simple algebra is the matrix ring of a division ring; thus, each similarity class is represented by a unique division ring. For example, Br(k) is trivial if k is a finite field or an algebraically closed field (more generally <a href="/wiki/Quasi-algebraically_closed_field" title="Quasi-algebraically closed field">quasi-algebraically closed field</a>; cf. <a href="/wiki/Tsen%27s_theorem" title="Tsen's theorem">Tsen's theorem</a>). <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Br} (\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Br</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Br} (\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c8afc731a9e2e565c2b768740c1ef3e457c9ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.045ex; height:2.843ex;" alt="{\displaystyle \operatorname {Br} (\mathbb {R} )}"> has order 2 (a special case of the <a href="/wiki/Frobenius_theorem_(real_division_algebras)" title="Frobenius theorem (real division algebras)">theorem of Frobenius</a>). Finally, if k is a nonarchimedean <a href="/wiki/Local_field" title="Local field">local field</a> (for example, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}">⁠), then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Br} (k)=\mathbb {Q} /\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Br</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Br} (k)=\mathbb {Q} /\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8693ac11b4cd77c6a9a0bd858d3dc9a1f8960787" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.197ex; height:2.843ex;" alt="{\displaystyle \operatorname {Br} (k)=\mathbb {Q} /\mathbb {Z} }"> through the <a href="/wiki/Hasse_invariant_of_an_algebra" title="Hasse invariant of an algebra">invariant map</a>. Now, if F is a field extension of k, then the base extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\otimes _{k}F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\otimes _{k}F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61c0ea46da2b0e8863daa48560c6ea9ae316da0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.478ex; height:2.509ex;" alt="{\displaystyle -\otimes _{k}F}"> induces Br(k) → Br(F). Its kernel is denoted by Br(F / k). It consists of [A] such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\otimes _{k}F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msub> <mo>⊗</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mi>F</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\otimes _{k}F}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b94ca94291b3216fd6f42f2624996c75350e583" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.413ex; height:2.509ex;" alt="{\displaystyle A\otimes _{k}F}"> is a matrix ring over F (that is, A is split by F.) If the extension is finite and Galois, then Br(F / k) is canonically isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{2}\left(\operatorname {Gal} (F/k),k^{*}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>Gal</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{2}\left(\operatorname {Gal} (F/k),k^{*}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f38858b5f35160ba7a1c07fc0a839895fb75755" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.245ex; height:3.176ex;" alt="{\displaystyle H^{2}\left(\operatorname {Gal} (F/k),k^{*}\right).}"><a href="#cite_note-FOOTNOTESerre1950-55">[50]</a> <a href="/wiki/Azumaya_algebra" title="Azumaya algebra">Azumaya algebras</a> generalize the notion of central simple algebras to a commutative local ring. <div class="mw-heading mw-heading3"><h3 id="Valuation_ring">Valuation ring</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=39" title="Edit section: Valuation ring">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Valuation_ring" title="Valuation ring">Valuation ring</a></div> If K is a field, a <a href="/wiki/Valuation_(algebra)" title="Valuation (algebra)">valuation</a> v is a group homomorphism from the multiplicative group K∗ to a totally ordered abelian group G such that, for any f, g in K with f + g nonzero, v(f + g) ≥ min{v(f), v(g)}. The <a href="/wiki/Valuation_ring" title="Valuation ring">valuation ring</a> of v is the subring of K consisting of zero and all nonzero f such that v(f) ≥ 0. Examples: <ul><li>The field of <a href="/wiki/Formal_Laurent_series" class="mw-redirect" title="Formal Laurent series">formal Laurent series</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(\!(t)\!)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">(</mo> <mspace width="negativethinmathspace" /> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="negativethinmathspace" /> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k(\!(t)\!)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3169860a904aa0d2eeed9dd7beb0bad569a1bce7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.895ex; height:2.843ex;" alt="{\displaystyle k(\!(t)\!)}"> over a field k comes with the valuation v such that v(f) is the least degree of a nonzero term in f; the valuation ring of v is the <a href="/wiki/Formal_power_series_ring" class="mw-redirect" title="Formal power series ring">formal power series ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k[\![t]\!].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">[</mo> <mspace width="negativethinmathspace" /> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mspace width="negativethinmathspace" /> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k[\![t]\!].}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db5e2887e456f95fb545b62435a54ac319e7a782" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.511ex; height:2.843ex;" alt="{\displaystyle k[\![t]\!].}"></li> <li>More generally, given a field k and a totally ordered abelian group G, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k(\!(G)\!)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo stretchy="false">(</mo> <mspace width="negativethinmathspace" /> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mspace width="negativethinmathspace" /> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k(\!(G)\!)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34ae08c40a0a9764ac588799a6c08ee8c896f2be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.882ex; height:2.843ex;" alt="{\displaystyle k(\!(G)\!)}"> be the set of all functions from G to k whose supports (the sets of points at which the functions are nonzero) are <a href="/wiki/Well_ordered" class="mw-redirect" title="Well ordered">well ordered</a>. It is a field with the multiplication given by <a href="/wiki/Convolution" title="Convolution">convolution</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(t)=\sum _{s\in G}f(s)g(t-s).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mo>∈</mo> <mi>G</mi> </mrow> </munder> <mi>f</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)(t)=\sum _{s\in G}f(s)g(t-s).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/350472eaccf731c5c0dfdce0a351e7f22829a1eb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:28.409ex; height:5.676ex;" alt="{\displaystyle (f*g)(t)=\sum _{s\in G}f(s)g(t-s).}"> It also comes with the valuation v such that v(f) is the least element in the support of f. The subring consisting of elements with finite support is called the <a href="/wiki/Group_ring" title="Group ring">group ring</a> of G (which makes sense even if G is not commutative). If G is the ring of integers, then we recover the previous example (by identifying f with the series whose nth coefficient is f(n).)</li></ul> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Novikov_ring" title="Novikov ring">Novikov ring</a> and <a href="/wiki/Uniserial_ring" class="mw-redirect" title="Uniserial ring">uniserial ring</a></div> <div class="mw-heading mw-heading2"><h2 id="Rings_with_extra_structure">Rings with extra structure</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=40" title="Edit section: Rings with extra structure">edit</a>]</div> A ring may be viewed as an <a href="/wiki/Abelian_group" title="Abelian group">abelian group</a> (by using the addition operation), with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example: <ul><li>An <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebra</a> is a ring that is also a <a href="/wiki/Vector_space" title="Vector space">vector space</a> over a field n such that the scalar multiplication is compatible with the ring multiplication. For instance, the set of n-by-n matrices over the real field ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">⁠ has dimension n2 as a real vector space.</li> <li>A ring R is a <a href="/wiki/Topological_ring" title="Topological ring">topological ring</a> if its set of elements R is given a <a href="/wiki/Topological_space" title="Topological space">topology</a> which makes the addition map (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +:R\times R\to R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mo>:</mo> <mi>R</mi> <mo>×</mo> <mi>R</mi> <mo stretchy="false">→</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +:R\times R\to R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4817b36ac24c9eca520832e9a231bf030eba826" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.492ex; height:2.343ex;" alt="{\displaystyle +:R\times R\to R}">) and the multiplication map ⋅ : R × R → R to be both <a href="/wiki/Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a> as maps between topological spaces (where X × X inherits the <a href="/wiki/Product_topology" title="Product topology">product topology</a> or any other product in the category). For example, n-by-n matrices over the real numbers could be given either the <a href="/wiki/Euclidean_topology" title="Euclidean topology">Euclidean topology</a>, or the <a href="/wiki/Zariski_topology" title="Zariski topology">Zariski topology</a>, and in either case one would obtain a topological ring.</li> <li>A <a href="/wiki/%CE%9B-ring" title="Λ-ring">λ-ring</a> is a commutative ring R together with operations λn: R → R that are like nth <a href="/wiki/Exterior_power" class="mw-redirect" title="Exterior power">exterior powers</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ^{n}(x+y)=\sum _{0}^{n}\lambda ^{i}(x)\lambda ^{n-i}(y).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>−</mo> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ^{n}(x+y)=\sum _{0}^{n}\lambda ^{i}(x)\lambda ^{n-i}(y).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb62cf6378a1dd0245379befa5b149a5de50705e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:29.874ex; height:6.843ex;" alt="{\displaystyle \lambda ^{n}(x+y)=\sum _{0}^{n}\lambda ^{i}(x)\lambda ^{n-i}(y).}"></dd></dl></li></ul> <dl><dd>For example, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">⁠ is a λ-ring with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ^{n}(x)={\binom {x}{n}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">(</mo> </mrow> <mfrac linethickness="0"> <mi>x</mi> <mi>n</mi> </mfrac> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">)</mo> </mrow> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ^{n}(x)={\binom {x}{n}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/294f88c6d54e36cc6b504bc1dbe6b0332cabd8fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.274ex; height:6.176ex;" alt="{\displaystyle \lambda ^{n}(x)={\binom {x}{n}},}"> the <a href="/wiki/Binomial_coefficient" title="Binomial coefficient">binomial coefficients</a>. The notion plays a central rule in the algebraic approach to the <a href="/wiki/Riemann%E2%80%93Roch_theorem" title="Riemann–Roch theorem">Riemann–Roch theorem</a>.</dd></dl> <ul><li>A <a href="/wiki/Totally_ordered_ring" class="mw-redirect" title="Totally ordered ring">totally ordered ring</a> is a ring with a <a href="/wiki/Total_ordering" class="mw-redirect" title="Total ordering">total ordering</a> that is compatible with ring operations.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Some_examples_of_the_ubiquity_of_rings">Some examples of the ubiquity of rings</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=41" title="Edit section: Some examples of the ubiquity of rings">edit</a>]</div> Many different kinds of <a href="/wiki/Mathematical_object" title="Mathematical object">mathematical objects</a> can be fruitfully analyzed in terms of some <a href="/wiki/Functor" title="Functor">associated ring</a>. <div class="mw-heading mw-heading3"><h3 id="Cohomology_ring_of_a_topological_space">Cohomology ring of a topological space</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=42" title="Edit section: Cohomology ring of a topological space">edit</a>]</div> To any <a href="/wiki/Topological_space" title="Topological space">topological space</a> X one can associate its integral <a href="/wiki/Cohomology_ring" title="Cohomology ring">cohomology ring</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{*}(X,\mathbb {Z} )=\bigoplus _{i=0}^{\infty }H^{i}(X,\mathbb {Z} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>⨁</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H^{*}(X,\mathbb {Z} )=\bigoplus _{i=0}^{\infty }H^{i}(X,\mathbb {Z} ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/007cfceac67b838abd3dc7413c8f5b3d26253776" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.451ex; height:6.843ex;" alt="{\displaystyle H^{*}(X,\mathbb {Z} )=\bigoplus _{i=0}^{\infty }H^{i}(X,\mathbb {Z} ),}"></dd></dl> a <a href="/wiki/Graded_ring" title="Graded ring">graded ring</a>. There are also <a href="/wiki/Homology_group" class="mw-redirect" title="Homology group">homology groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{i}(X,\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{i}(X,\mathbb {Z} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27bfcca13ba463426618f76328f9b24346afb2f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.104ex; height:2.843ex;" alt="{\displaystyle H_{i}(X,\mathbb {Z} )}"> of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the <a href="/wiki/Sphere" title="Sphere">spheres</a> and <a href="/wiki/Torus" title="Torus">tori</a>, for which the methods of <a href="/wiki/Point-set_topology" class="mw-redirect" title="Point-set topology">point-set topology</a> are not well-suited. <a href="/wiki/Cohomology_group" class="mw-redirect" title="Cohomology group">Cohomology groups</a> were later defined in terms of homology groups in a way which is roughly analogous to the dual of a <a href="/wiki/Vector_space" title="Vector space">vector space</a>. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the <a href="/wiki/Universal_coefficient_theorem" title="Universal coefficient theorem">universal coefficient theorem</a>. However, the advantage of the cohomology groups is that there is a <a href="/wiki/Cup_product" title="Cup product">natural product</a>, which is analogous to the observation that one can multiply pointwise a k-<a href="/wiki/Multilinear_form" title="Multilinear form">multilinear form</a> and an l-multilinear form to get a (k + l)-multilinear form. The ring structure in cohomology provides the foundation for <a href="/wiki/Characteristic_class" title="Characteristic class">characteristic classes</a> of <a href="/wiki/Fiber_bundle" title="Fiber bundle">fiber bundles</a>, intersection theory on manifolds and <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic varieties</a>, <a href="/wiki/Schubert_calculus" title="Schubert calculus">Schubert calculus</a> and much more. <div class="mw-heading mw-heading3"><h3 id="Burnside_ring_of_a_group">Burnside ring of a group</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=43" title="Edit section: Burnside ring of a group">edit</a>]</div> To any <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> is associated its <a href="/wiki/Burnside_ring" title="Burnside ring">Burnside ring</a> which uses a ring to describe the various ways the group can <a href="/wiki/Group_action_(mathematics)" class="mw-redirect" title="Group action (mathematics)">act</a> on a finite set. The Burnside ring's additive group is the <a href="/wiki/Free_abelian_group" title="Free abelian group">free abelian group</a> whose basis is the set of transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the <a href="/wiki/Representation_ring" title="Representation ring">representation ring</a>: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers. <div class="mw-heading mw-heading3"><h3 id="Representation_ring_of_a_group_ring">Representation ring of a group ring</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=44" title="Edit section: Representation ring of a group ring">edit</a>]</div> To any <a href="/wiki/Group_ring" title="Group ring">group ring</a> or <a href="/wiki/Hopf_algebra" title="Hopf algebra">Hopf algebra</a> is associated its <a href="/wiki/Representation_ring" title="Representation ring">representation ring</a> or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from <a href="/wiki/Character_theory" title="Character theory">character theory</a>, which is more or less the <a href="/wiki/Grothendieck_group" title="Grothendieck group">Grothendieck group</a> given a ring structure. <div class="mw-heading mw-heading3"><h3 id="Function_field_of_an_irreducible_algebraic_variety">Function field of an irreducible algebraic variety</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=45" title="Edit section: Function field of an irreducible algebraic variety">edit</a>]</div> To any irreducible <a href="/wiki/Algebraic_variety" title="Algebraic variety">algebraic variety</a> is associated its <a href="/wiki/Function_field_of_an_algebraic_variety" title="Function field of an algebraic variety">function field</a>. The points of an algebraic variety correspond to <a href="/wiki/Valuation_ring" title="Valuation ring">valuation rings</a> contained in the function field and containing the <a href="/wiki/Coordinate_ring" class="mw-redirect" title="Coordinate ring">coordinate ring</a>. The study of <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a> makes heavy use of <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a> to study geometric concepts in terms of ring-theoretic properties. <a href="/wiki/Birational_geometry" title="Birational geometry">Birational geometry</a> studies maps between the subrings of the function field. <div class="mw-heading mw-heading3"><h3 id="Face_ring_of_a_simplicial_complex">Face ring of a simplicial complex</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=46" title="Edit section: Face ring of a simplicial complex">edit</a>]</div> Every <a href="/wiki/Simplicial_complex" title="Simplicial complex">simplicial complex</a> has an associated face ring, also called its <a href="/wiki/Stanley%E2%80%93Reisner_ring" title="Stanley–Reisner ring">Stanley–Reisner ring</a>. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in <a href="/wiki/Algebraic_combinatorics" title="Algebraic combinatorics">algebraic combinatorics</a>. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of <a href="/wiki/Simplicial_polytope" title="Simplicial polytope">simplicial polytopes</a>. <div class="mw-heading mw-heading2"><h2 id="Category-theoretic_description">Category-theoretic description</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=47" title="Edit section: Category-theoretic description">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Category_of_rings" title="Category of rings">Category of rings</a></div> Every ring can be thought of as a <a href="/wiki/Monoid_(category_theory)" title="Monoid (category theory)">monoid</a> in Ab, the <a href="/wiki/Category_of_abelian_groups" title="Category of abelian groups">category of abelian groups</a> (thought of as a <a href="/wiki/Monoidal_category" title="Monoidal category">monoidal category</a> under the <a href="/wiki/Tensor_product_of_abelian_groups" class="mw-redirect" title="Tensor product of abelian groups">tensor product of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">⁠-modules</a>). The monoid action of a ring R on an abelian group is simply an <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">R-module</a>. Essentially, an R-module is a generalization of the notion of a <a href="/wiki/Vector_space" title="Vector space">vector space</a> – where rather than a vector space over a field, one has a "vector space over a ring". Let (A, +) be an abelian group and let End(A) be its <a href="/wiki/Endomorphism_ring" title="Endomorphism ring">endomorphism ring</a> (see above). Note that, essentially, End(A) is the set of all morphisms of A, where if f is in End(A), and g is in End(A), the following rules may be used to compute f + g and f ⋅ g: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&(f+g)(x)=f(x)+g(x)\\&(f\cdot g)(x)=f(g(x)),\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>+</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>⋅</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&(f+g)(x)=f(x)+g(x)\\&(f\cdot g)(x)=f(g(x)),\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a11be4f8f2849ef871ab5aa11de39757a7d63130" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.546ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}&(f+g)(x)=f(x)+g(x)\\&(f\cdot g)(x)=f(g(x)),\end{aligned}}}"></dd></dl> where + as in f(x) + g(x) is addition in A, and function composition is denoted from right to left. Therefore, <a href="/wiki/Functor" title="Functor">associated</a> to any abelian group, is a ring. Conversely, given any ring, (R, +, ⋅ ), (R, +) is an abelian group. Furthermore, for every r in R, right (or left) multiplication by r gives rise to a morphism of (R, +), by right (or left) distributivity. Let A = (R, +). Consider those <a href="/wiki/Endomorphism" title="Endomorphism">endomorphisms</a> of A, that "factor through" right (or left) multiplication of R. In other words, let EndR(A) be the set of all morphisms m of A, having the property that m(r ⋅ x) = r ⋅ m(x). It was seen that every r in R gives rise to a morphism of A: right multiplication by r. It is in fact true that this association of any element of R, to a morphism of A, as a function from R to EndR(A), is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian X-group (by X-group, it is meant a group with X being its <a href="/wiki/Group_with_operators" title="Group with operators">set of operators</a>).<a href="#cite_note-FOOTNOTEJacobson2009162Theorem_3.2-56">[51]</a> In essence, the most general form of a ring, is the endomorphism group of some abelian X-group. Any ring can be seen as a <a href="/wiki/Preadditive_category" title="Preadditive category">preadditive category</a> with a single object. It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context. <a href="/wiki/Additive_functor" class="mw-redirect" title="Additive functor">Additive functors</a> between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of <a href="/wiki/Morphism" title="Morphism">morphisms</a> closed under addition and under composition with arbitrary morphisms. <div class="mw-heading mw-heading2"><h2 id="Generalization">Generalization</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=48" title="Edit section: Generalization">edit</a>]</div> Algebraists have defined structures more general than rings by weakening or dropping some of ring axioms. <div class="mw-heading mw-heading3"><h3 id="Rng">Rng</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=49" title="Edit section: Rng">edit</a>]</div> A <a href="/wiki/Rng_(algebra)" title="Rng (algebra)">rng</a> is the same as a ring, except that the existence of a multiplicative identity is not assumed.<a href="#cite_note-FOOTNOTEJacobson2009-57">[52]</a> <div class="mw-heading mw-heading3"><h3 id="Nonassociative_ring">Nonassociative ring</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=50" title="Edit section: Nonassociative ring">edit</a>]</div> A <a href="/wiki/Nonassociative_ring" class="mw-redirect" title="Nonassociative ring">nonassociative ring</a> is an algebraic structure that satisfies all of the ring axioms except the associative property and the existence of a multiplicative identity. A notable example is a <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a>. There exists some structure theory for such algebras that generalizes the analogous results for Lie algebras and associative algebras.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <div class="mw-heading mw-heading3"><h3 id="Semiring">Semiring</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=51" title="Edit section: Semiring">edit</a>]</div> A <a href="/wiki/Semiring" title="Semiring">semiring</a> (sometimes rig) is obtained by weakening the assumption that (R, +) is an abelian group to the assumption that (R, +) is a commutative monoid, and adding the axiom that 0 ⋅ a = a ⋅ 0 = 0 for all a in R (since it no longer follows from the other axioms). Examples: <ul><li>the non-negative integers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,1,2,\ldots \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,1,2,\ldots \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb18e9a1a9837670a4823e5b3658cd3aa8cc8e88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.637ex; height:2.843ex;" alt="{\displaystyle \{0,1,2,\ldots \}}"> with ordinary addition and multiplication;</li> <li>the <a href="/wiki/Tropical_semiring" title="Tropical semiring">tropical semiring</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Other_ring-like_objects">Other ring-like objects</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=52" title="Edit section: Other ring-like objects">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Ring_object_in_a_category">Ring object in a category</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=53" title="Edit section: Ring object in a category">edit</a>]</div> Let C be a category with finite <a href="/wiki/Product_(category_theory)" title="Product (category theory)">products</a>. Let pt denote a <a href="/wiki/Terminal_object" class="mw-redirect" title="Terminal object">terminal object</a> of C (an empty product). A ring object in C is an object R equipped with morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\times R\;{\stackrel {a}{\to }}\,R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>×</mo> <mi>R</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\times R\;{\stackrel {a}{\to }}\,R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3de6e0646f13dd9b1133145e4563fbf7e20b6dbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.488ex; height:3.176ex;" alt="{\displaystyle R\times R\;{\stackrel {a}{\to }}\,R}"> (addition), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\times R\;{\stackrel {m}{\to }}\,R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>×</mo> <mi>R</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\times R\;{\stackrel {m}{\to }}\,R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d87e6f5625947f205990dcd191ecdc9f614fc48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.488ex; height:3.176ex;" alt="{\displaystyle R\times R\;{\stackrel {m}{\to }}\,R}"> (multiplication), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {pt} {\stackrel {0}{\to }}\,R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>pt</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {pt} {\stackrel {0}{\to }}\,R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd7ddef0dcb0b4f63601addbbba841818aca8905" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.059ex; height:3.843ex;" alt="{\displaystyle \operatorname {pt} {\stackrel {0}{\to }}\,R}"> (additive identity), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\;{\stackrel {i}{\to }}\,R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\;{\stackrel {i}{\to }}\,R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f756799ecd6928776d7a00c4a45a1ce971bf1e7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.884ex; height:3.509ex;" alt="{\displaystyle R\;{\stackrel {i}{\to }}\,R}"> (additive inverse), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {pt} {\stackrel {1}{\to }}\,R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>pt</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">→</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </mover> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {pt} {\stackrel {1}{\to }}\,R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02f1a2fe161c516d9adac5334883dd04c9c3e09c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.059ex; height:3.843ex;" alt="{\displaystyle \operatorname {pt} {\stackrel {1}{\to }}\,R}"> (multiplicative identity) satisfying the usual ring axioms. Equivalently, a ring object is an object R equipped with a factorization of its functor of points <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{R}=\operatorname {Hom} (-,R):C^{\operatorname {op} }\to \mathbf {Sets} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>=</mo> <mi>Hom</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>:</mo> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>op</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> <mi mathvariant="bold">e</mi> <mi mathvariant="bold">t</mi> <mi mathvariant="bold">s</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{R}=\operatorname {Hom} (-,R):C^{\operatorname {op} }\to \mathbf {Sets} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f8d69b8d4c90ba7492d015b23e93831a4403059" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.297ex; height:2.843ex;" alt="{\displaystyle h_{R}=\operatorname {Hom} (-,R):C^{\operatorname {op} }\to \mathbf {Sets} }"> through the category of rings: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C^{\operatorname {op} }\to \mathbf {Rings} {\stackrel {\textrm {forgetful}}{\longrightarrow }}\mathbf {Sets} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>op</mi> </mrow> </msup> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">R</mi> <mi mathvariant="bold">i</mi> <mi mathvariant="bold">n</mi> <mi mathvariant="bold">g</mi> <mi mathvariant="bold">s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-REL"> <mover> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <mo stretchy="false">⟶</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>forgetful</mtext> </mrow> </mrow> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> <mi mathvariant="bold">e</mi> <mi mathvariant="bold">t</mi> <mi mathvariant="bold">s</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C^{\operatorname {op} }\to \mathbf {Rings} {\stackrel {\textrm {forgetful}}{\longrightarrow }}\mathbf {Sets} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcb39c4a8d8364f343dce7ddf4103e29084cb2d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.492ex; height:4.176ex;" alt="{\displaystyle C^{\operatorname {op} }\to \mathbf {Rings} {\stackrel {\textrm {forgetful}}{\longrightarrow }}\mathbf {Sets} .}"> <div class="mw-heading mw-heading3"><h3 id="Ring_scheme">Ring scheme</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=54" title="Edit section: Ring scheme">edit</a>]</div> In algebraic geometry, a ring scheme over a base <a href="/wiki/Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> S is a ring object in the category of S-schemes. One example is the ring scheme Wn over ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Spec} \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Spec</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Spec} \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd34331bea4cfc6371708f3e886e16c8c7adedcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.587ex; height:2.509ex;" alt="{\displaystyle \operatorname {Spec} \mathbb {Z} }">⁠, which for any commutative ring A returns the ring Wn(A) of p-isotypic <a href="/wiki/Witt_vector" title="Witt vector">Witt vectors</a> of length n over A.<a href="#cite_note-58">[53]</a> <div class="mw-heading mw-heading3"><h3 id="Ring_spectrum">Ring spectrum</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=55" title="Edit section: Ring spectrum">edit</a>]</div> In <a href="/wiki/Algebraic_topology" title="Algebraic topology">algebraic topology</a>, a <a href="/wiki/Ring_spectrum" title="Ring spectrum">ring spectrum</a> is a <a href="/wiki/Spectrum_(topology)" title="Spectrum (topology)">spectrum</a> X together with a multiplication <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu :X\wedge X\to X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>:</mo> <mi>X</mi> <mo>∧</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu :X\wedge X\to X}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd313f336cfb91f1fbc73e1ece962fc9d7f81cae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.476ex; height:2.676ex;" alt="{\displaystyle \mu :X\wedge X\to X}"> and a unit map S → X from the <a href="/wiki/Sphere_spectrum" title="Sphere spectrum">sphere spectrum</a> S, such that the ring axiom diagrams commute up to homotopy. In practice, it is common to define a ring spectrum as a <a href="/wiki/Monoid_object" class="mw-redirect" title="Monoid object">monoid object</a> in a good category of spectra such as the category of <a href="/wiki/Symmetric_spectrum" title="Symmetric spectrum">symmetric spectra</a>. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=56" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></div> <div class="side-box-text plainlist">Wikibooks has a book on the topic of: <a href="https://en.wikibooks.org/wiki/Abstract_Algebra/Rings" class="extiw" title="wikibooks:Abstract Algebra/Rings">Abstract Algebra/Rings</a></div></div> </div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 18em;"> <ul><li><a href="/wiki/Algebra_over_a_commutative_ring" class="mw-redirect" title="Algebra over a commutative ring">Algebra over a commutative ring</a></li> <li><a href="/wiki/Categorical_ring" class="mw-redirect" title="Categorical ring">Categorical ring</a></li> <li><a href="/wiki/Category_of_rings" title="Category of rings">Category of rings</a></li> <li><a href="/wiki/Glossary_of_ring_theory" title="Glossary of ring theory">Glossary of ring theory</a></li> <li><a href="/wiki/Non-associative_algebra" title="Non-associative algebra">Non-associative algebra</a></li> <li><a href="/wiki/Ring_of_sets" title="Ring of sets">Ring of sets</a></li> <li><a href="/wiki/Semiring" title="Semiring">Semiring</a></li> <li><a href="/wiki/Spectrum_of_a_ring" title="Spectrum of a ring">Spectrum of a ring</a></li> <li><a href="/wiki/Simplicial_commutative_ring" title="Simplicial commutative ring">Simplicial commutative ring</a></li></ul> </div> Special types of rings: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1184024115"><div class="div-col" style="column-width: 22em;"> <ul><li><a href="/wiki/Boolean_ring" title="Boolean ring">Boolean ring</a></li> <li><a href="/wiki/Dedekind_ring" class="mw-redirect" title="Dedekind ring">Dedekind ring</a></li> <li><a href="/wiki/Differential_ring" class="mw-redirect" title="Differential ring">Differential ring</a></li> <li><a href="/wiki/Exponential_field" title="Exponential field">Exponential ring</a></li> <li><a href="/wiki/Finite_ring" title="Finite ring">Finite ring</a></li> <li><a href="/wiki/Lie_ring" class="mw-redirect" title="Lie ring">Lie ring</a></li> <li><a href="/wiki/Local_ring" title="Local ring">Local ring</a></li> <li><a href="/wiki/Noetherian_ring" title="Noetherian ring">Noetherian</a> and <a href="/wiki/Artinian_ring" title="Artinian ring">artinian rings</a></li> <li><a href="/wiki/Ordered_ring" title="Ordered ring">Ordered ring</a></li> <li><a href="/wiki/Poisson_ring" title="Poisson ring">Poisson ring</a></li> <li><a href="/wiki/Reduced_ring" title="Reduced ring">Reduced ring</a></li> <li><a href="/wiki/Regular_ring" class="mw-redirect" title="Regular ring">Regular ring</a></li> <li><a href="/wiki/Ring_of_periods" class="mw-redirect" title="Ring of periods">Ring of periods</a></li> <li><a href="/wiki/SBI_ring" title="SBI ring">SBI ring</a></li> <li><a href="/wiki/Valuation_ring" title="Valuation ring">Valuation ring</a> and <a href="/wiki/Discrete_valuation_ring" title="Discrete valuation ring">discrete valuation ring</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=57" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> This means that each operation is defined and produces a unique result in R for each ordered pair of elements of R. </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> The existence of 1 is not assumed by some authors; here, the term <a href="/wiki/Rng_(algebra)" title="Rng (algebra)">rng</a> is used if existence of a multiplicative identity is not assumed. See <a class="mw-selflink-fragment" href="#Variations_on_the_definition">next subsection</a>. </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> Poonen claims that "the natural extension of associativity demands that rings should contain an empty product, so it is natural to require rings to have a 1". </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> Some other authors such as Lang further require a zero divisor to be nonzero. </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> Such a central idempotent is called <a href="/wiki/Centrally_primitive" class="mw-redirect" title="Centrally primitive">centrally primitive</a>. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Citations">Citations</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=58" title="Edit section: Citations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-FOOTNOTEBourbaki198996Ch_1,_§8.1-2"><a href="#cite_ref-FOOTNOTEBourbaki198996Ch_1,_§8.1_2-0">^</a> <a href="#CITEREFBourbaki1989">Bourbaki (1989)</a>, p. 96, Ch 1, §8.1 </li> <li id="cite_note-FOOTNOTEMac_LaneBirkhoff196785-3"><a href="#cite_ref-FOOTNOTEMac_LaneBirkhoff196785_3-0">^</a> <a href="#CITEREFMac_LaneBirkhoff1967">Mac Lane & Birkhoff (1967)</a>, p. 85 </li> <li id="cite_note-FOOTNOTELang200283-4">^ <a href="#cite_ref-FOOTNOTELang200283_4-0">a</a> <a href="#cite_ref-FOOTNOTELang200283_4-1">b</a> <a href="#CITEREFLang2002">Lang (2002)</a>, p. 83 </li> <li id="cite_note-FOOTNOTEIsaacs1994160-6"><a href="#cite_ref-FOOTNOTEIsaacs1994160_6-0">^</a> <a href="#CITEREFIsaacs1994">Isaacs (1994)</a>, p. 160 </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Non-associative_rings_and_algebras">"Non-associative rings and algebras"</a>. Encyclopedia of Mathematics.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Non-associative+rings+and+algebras&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%2FNon-associative_rings_and_algebras&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-FOOTNOTEIsaacs1994161-8"><a href="#cite_ref-FOOTNOTEIsaacs1994161_8-0">^</a> <a href="#CITEREFIsaacs1994">Isaacs (1994)</a>, p. 161 </li> <li id="cite_note-FOOTNOTELam2001Theorem_3.1-9"><a href="#cite_ref-FOOTNOTELam2001Theorem_3.1_9-0">^</a> <a href="#CITEREFLam2001">Lam (2001)</a>, Theorem 3.1 </li> <li id="cite_note-FOOTNOTELang2005Ch_V,_§3-10"><a href="#cite_ref-FOOTNOTELang2005Ch_V,_§3_10-0">^</a> <a href="#CITEREFLang2005">Lang (2005)</a>, Ch V, §3. </li> <li id="cite_note-FOOTNOTESerre20063-11"><a href="#cite_ref-FOOTNOTESerre20063_11-0">^</a> <a href="#CITEREFSerre2006">Serre (2006)</a>, p. 3 </li> <li id="cite_note-FOOTNOTESerre1979158-12"><a href="#cite_ref-FOOTNOTESerre1979158_12-0">^</a> <a href="#CITEREFSerre1979">Serre (1979)</a>, p. 158 </li> <li id="cite_note-history-13"><a href="#cite_ref-history_13-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathshistory.st-andrews.ac.uk/HistTopics/Ring_theory/">"The development of Ring Theory"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+development+of+Ring+Theory&rft_id=https%3A%2F%2Fmathshistory.st-andrews.ac.uk%2FHistTopics%2FRing_theory%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-FOOTNOTEKleiner199827-14"><a href="#cite_ref-FOOTNOTEKleiner199827_14-0">^</a> <a href="#CITEREFKleiner1998">Kleiner (1998)</a>, p. 27 </li> <li id="cite_note-FOOTNOTEHilbert1897-15"><a href="#cite_ref-FOOTNOTEHilbert1897_15-0">^</a> <a href="#CITEREFHilbert1897">Hilbert (1897)</a> </li> <li id="cite_note-FOOTNOTECohn1980[httpsarchiveorgdetailsadvancednumberth00cohn_0page49_p._49]-16"><a href="#cite_ref-FOOTNOTECohn1980[httpsarchiveorgdetailsadvancednumberth00cohn_0page49_p._49]_16-0">^</a> <a href="#CITEREFCohn1980">Cohn (1980)</a>, <a rel="nofollow" class="external text" href="https://archive.org/details/advancednumberth00cohn_0/page/49">p. 49</a> </li> <li id="cite_note-FOOTNOTEFraenkel1915143–145-17"><a href="#cite_ref-FOOTNOTEFraenkel1915143–145_17-0">^</a> <a href="#CITEREFFraenkel1915">Fraenkel (1915)</a>, pp. 143–145 </li> <li id="cite_note-FOOTNOTEJacobson200986footnote_1-18"><a href="#cite_ref-FOOTNOTEJacobson200986footnote_1_18-0">^</a> <a href="#CITEREFJacobson2009">Jacobson (2009)</a>, p. 86, footnote 1 </li> <li id="cite_note-FOOTNOTEFraenkel1915144axiom_''R''8)-19"><a href="#cite_ref-FOOTNOTEFraenkel1915144axiom_''R''8)_19-0">^</a> <a href="#CITEREFFraenkel1915">Fraenkel (1915)</a>, p. 144, axiom R8) </li> <li id="cite_note-FOOTNOTENoether192129-20">^ <a href="#cite_ref-FOOTNOTENoether192129_20-0">a</a> <a href="#cite_ref-FOOTNOTENoether192129_20-1">b</a> <a href="#CITEREFNoether1921">Noether (1921)</a>, p. 29 </li> <li id="cite_note-FOOTNOTEFraenkel1915144axiom_''R''7)-21"><a href="#cite_ref-FOOTNOTEFraenkel1915144axiom_''R''7)_21-0">^</a> <a href="#CITEREFFraenkel1915">Fraenkel (1915)</a>, p. 144, axiom R7) </li> <li id="cite_note-FOOTNOTEvan_der_Waerden1930-22"><a href="#cite_ref-FOOTNOTEvan_der_Waerden1930_22-0">^</a> <a href="#CITEREFvan_der_Waerden1930">van der Waerden (1930)</a> </li> <li id="cite_note-FOOTNOTEZariskiSamuel1958-23"><a href="#cite_ref-FOOTNOTEZariskiSamuel1958_23-0">^</a> <a href="#CITEREFZariskiSamuel1958">Zariski & Samuel (1958)</a> </li> <li id="cite_note-FOOTNOTEArtin2018346-24"><a href="#cite_ref-FOOTNOTEArtin2018346_24-0">^</a> <a href="#CITEREFArtin2018">Artin (2018)</a>, p. 346 </li> <li id="cite_note-FOOTNOTEBourbaki198996-25"><a href="#cite_ref-FOOTNOTEBourbaki198996_25-0">^</a> <a href="#CITEREFBourbaki1989">Bourbaki (1989)</a>, p. 96 </li> <li id="cite_note-FOOTNOTEEisenbud199511-26"><a href="#cite_ref-FOOTNOTEEisenbud199511_26-0">^</a> <a href="#CITEREFEisenbud1995">Eisenbud (1995)</a>, p. 11 </li> <li id="cite_note-FOOTNOTEGallian2006235-27"><a href="#cite_ref-FOOTNOTEGallian2006235_27-0">^</a> <a href="#CITEREFGallian2006">Gallian (2006)</a>, p. 235 </li> <li id="cite_note-FOOTNOTEHungerford199742-28"><a href="#cite_ref-FOOTNOTEHungerford199742_28-0">^</a> <a href="#CITEREFHungerford1997">Hungerford (1997)</a>, p. 42 </li> <li id="cite_note-FOOTNOTEWarner1965188-29"><a href="#cite_ref-FOOTNOTEWarner1965188_29-0">^</a> <a href="#CITEREFWarner1965">Warner (1965)</a>, p. 188 </li> <li id="cite_note-FOOTNOTEGarling2022-30"><a href="#cite_ref-FOOTNOTEGarling2022_30-0">^</a> <a href="#CITEREFGarling2022">Garling (2022)</a> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php/Associative_rings_and_algebras">"Associative rings and algebras"</a>. Encyclopedia of Mathematics.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Associative+rings+and+algebras&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%2FAssociative_rings_and_algebras&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"> </li> <li id="cite_note-FOOTNOTEGardnerWiegandt2003-32"><a href="#cite_ref-FOOTNOTEGardnerWiegandt2003_32-0">^</a> <a href="#CITEREFGardnerWiegandt2003">Gardner & Wiegandt (2003)</a> </li> <li id="cite_note-FOOTNOTEPoonen2019-34"><a href="#cite_ref-FOOTNOTEPoonen2019_34-0">^</a> <a href="#CITEREFPoonen2019">Poonen (2019)</a> </li> <li id="cite_note-FOOTNOTEWilder1965176-35"><a href="#cite_ref-FOOTNOTEWilder1965176_35-0">^</a> <a href="#CITEREFWilder1965">Wilder (1965)</a>, p. 176 </li> <li id="cite_note-FOOTNOTERotman19987-36"><a href="#cite_ref-FOOTNOTERotman19987_36-0">^</a> <a href="#CITEREFRotman1998">Rotman (1998)</a>, p. 7 </li> <li id="cite_note-FOOTNOTEJacobson2009155-37"><a href="#cite_ref-FOOTNOTEJacobson2009155_37-0">^</a> <a href="#CITEREFJacobson2009">Jacobson (2009)</a>, p. 155 </li> <li id="cite_note-FOOTNOTEBourbaki198998-38"><a href="#cite_ref-FOOTNOTEBourbaki198998_38-0">^</a> <a href="#CITEREFBourbaki1989">Bourbaki (1989)</a>, p. 98 </li> <li id="cite_note-FOOTNOTECohn2003Theorem_4.5.1-40"><a href="#cite_ref-FOOTNOTECohn2003Theorem_4.5.1_40-0">^</a> <a href="#CITEREFCohn2003">Cohn (2003)</a>, Theorem 4.5.1 </li> <li id="cite_note-FOOTNOTEJacobson2009122Theorem_2.10-42"><a href="#cite_ref-FOOTNOTEJacobson2009122Theorem_2.10_42-0">^</a> <a href="#CITEREFJacobson2009">Jacobson (2009)</a>, p. 122, Theorem 2.10 </li> <li id="cite_note-FOOTNOTEBourbaki1964Ch_5._§1,_Lemma_2-43"><a href="#cite_ref-FOOTNOTEBourbaki1964Ch_5._§1,_Lemma_2_43-0">^</a> <a href="#CITEREFBourbaki1964">Bourbaki (1964)</a>, Ch 5. §1, Lemma 2 </li> <li id="cite_note-FOOTNOTECohn20034.4-44">^ <a href="#cite_ref-FOOTNOTECohn20034.4_44-0">a</a> <a href="#cite_ref-FOOTNOTECohn20034.4_44-1">b</a> <a href="#CITEREFCohn2003">Cohn (2003)</a>, 4.4 </li> <li id="cite_note-FOOTNOTELang2002Ch._XVII._Proposition_1.1-45"><a href="#cite_ref-FOOTNOTELang2002Ch._XVII._Proposition_1.1_45-0">^</a> <a href="#CITEREFLang2002">Lang (2002)</a>, Ch. XVII. Proposition 1.1 </li> <li id="cite_note-FOOTNOTECohn1995Proposition_1.3.1-46"><a href="#cite_ref-FOOTNOTECohn1995Proposition_1.3.1_46-0">^</a> <a href="#CITEREFCohn1995">Cohn (1995)</a>, Proposition 1.3.1 </li> <li id="cite_note-FOOTNOTEEisenbud1995Exercise_2.2-47"><a href="#cite_ref-FOOTNOTEEisenbud1995Exercise_2.2_47-0">^</a> <a href="#CITEREFEisenbud1995">Eisenbud (1995)</a>, Exercise 2.2 </li> <li id="cite_note-FOOTNOTEMilne2012Proposition_6.4-48"><a href="#cite_ref-FOOTNOTEMilne2012Proposition_6.4_48-0">^</a> <a href="#CITEREFMilne2012">Milne (2012)</a>, Proposition 6.4 </li> <li id="cite_note-FOOTNOTEMilne2012end_of_Chapter_7-49"><a href="#cite_ref-FOOTNOTEMilne2012end_of_Chapter_7_49-0">^</a> <a href="#CITEREFMilne2012">Milne (2012)</a>, end of Chapter 7 </li> <li id="cite_note-FOOTNOTEAtiyahMacdonald1969Theorem_10.17_and_its_corollaries-50"><a href="#cite_ref-FOOTNOTEAtiyahMacdonald1969Theorem_10.17_and_its_corollaries_50-0">^</a> <a href="#CITEREFAtiyahMacdonald1969">Atiyah & Macdonald (1969)</a>, Theorem 10.17 and its corollaries </li> <li id="cite_note-FOOTNOTECohn1995[httpsbooksgooglecombooksidu-4ADgUgpSMCpgPA242_pg._242]-51"><a href="#cite_ref-FOOTNOTECohn1995[httpsbooksgooglecombooksidu-4ADgUgpSMCpgPA242_pg._242]_51-0">^</a> <a href="#CITEREFCohn1995">Cohn (1995)</a>, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=u-4ADgUgpSMC&pg=PA242">pg. 242</a> </li> <li id="cite_note-FOOTNOTELang2002Ch_XIV,_§2-52"><a href="#cite_ref-FOOTNOTELang2002Ch_XIV,_§2_52-0">^</a> <a href="#CITEREFLang2002">Lang (2002)</a>, Ch XIV, §2 </li> <li id="cite_note-FOOTNOTEWeibel2013[httpsbooksgooglecombooksidJa8xAAAAQBAJpgPA26_26]Ch_1,_Theorem_3.8-53"><a href="#cite_ref-FOOTNOTEWeibel2013[httpsbooksgooglecombooksidJa8xAAAAQBAJpgPA26_26]Ch_1,_Theorem_3.8_53-0">^</a> <a href="#CITEREFWeibel2013">Weibel (2013)</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ja8xAAAAQBAJ&pg=PA26">26</a>, Ch 1, Theorem 3.8 </li> <li id="cite_note-FOOTNOTEMilneCFTCh_IV,_§2-54"><a href="#cite_ref-FOOTNOTEMilneCFTCh_IV,_§2_54-0">^</a> <a href="#CITEREFMilneCFT">Milne & CFT</a>, Ch IV, §2 </li> <li id="cite_note-FOOTNOTESerre1950-55"><a href="#cite_ref-FOOTNOTESerre1950_55-0">^</a> <a href="#CITEREFSerre1950">Serre (1950)</a> </li> <li id="cite_note-FOOTNOTEJacobson2009162Theorem_3.2-56"><a href="#cite_ref-FOOTNOTEJacobson2009162Theorem_3.2_56-0">^</a> <a href="#CITEREFJacobson2009">Jacobson (2009)</a>, p. 162, Theorem 3.2 </li> <li id="cite_note-FOOTNOTEJacobson2009-57"><a href="#cite_ref-FOOTNOTEJacobson2009_57-0">^</a> <a href="#CITEREFJacobson2009">Jacobson (2009)</a> </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> Serre, p. 44 </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=59" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGarling2022" class="citation book cs1">Garling, D. J. H. (2022). Galois Theory and Its Algebraic Background (2nd ed.). Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-108-83892-4" title="Special:BookSources/978-1-108-83892-4"><bdi>978-1-108-83892-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Galois+Theory+and+Its+Algebraic+Background&rft.place=Cambridge&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=2022&rft.isbn=978-1-108-83892-4&rft.aulast=Garling&rft.aufirst=D.+J.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohn1980" class="citation cs2">Cohn, Harvey (1980), Advanced Number Theory, New York: Dover Publications, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-64023-5" title="Special:BookSources/978-0-486-64023-5"><bdi>978-0-486-64023-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Advanced+Number+Theory&rft.place=New+York&rft.pub=Dover+Publications&rft.date=1980&rft.isbn=978-0-486-64023-5&rft.aulast=Cohn&rft.aufirst=Harvey&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerre1950" class="citation cs2">Serre, J-P. (1950), <a rel="nofollow" class="external text" href="http://www.numdam.org/numdam-bin/feuilleter?id=SHC_1950-1951__3_">Applications algébriques de la cohomologie des groupes, I, II, Séminaire Henri Cartan, 1950/51</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Applications+alg%C3%A9briques+de+la+cohomologie+des+groupes%2C+I%2C+II%2C+S%C3%A9minaire+Henri+Cartan%2C+1950%2F51&rft.date=1950&rft.aulast=Serre&rft.aufirst=J-P.&rft_id=http%3A%2F%2Fwww.numdam.org%2Fnumdam-bin%2Ffeuilleter%3Fid%3DSHC_1950-1951__3_&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerre2006" class="citation cs2">Serre (2006), Lie algebras and Lie groups (2nd ed.), Springer</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lie+algebras+and+Lie+groups&rft.edition=2nd&rft.pub=Springer&rft.date=2006&rft.au=Serre&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"> [corrected 5th printing]</li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="General_references">General references</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=60" title="Edit section: General references">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArtin2018" class="citation book cs1"><a href="/wiki/Michael_Artin" title="Michael Artin">Artin, Michael</a> (2018). Algebra (2nd ed.). Pearson.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.edition=2nd&rft.pub=Pearson&rft.date=2018&rft.aulast=Artin&rft.aufirst=Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAtiyahMacdonald1969" class="citation book cs1"><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah, Michael</a>; <a href="/wiki/Ian_G._Macdonald" title="Ian G. Macdonald">Macdonald, Ian G.</a> (1969). Introduction to commutative algebra. Addison–Wesley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+commutative+algebra&rft.pub=Addison%E2%80%93Wesley&rft.date=1969&rft.aulast=Atiyah&rft.aufirst=Michael&rft.au=Macdonald%2C+Ian+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1964" class="citation book cs1"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, N.</a> (1964). Algèbre commutative. Hermann.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Alg%C3%A8bre+commutative&rft.pub=Hermann&rft.date=1964&rft.aulast=Bourbaki&rft.aufirst=N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBourbaki1989" class="citation book cs1"><a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki, N.</a> (1989). Algebra I, Chapters 1–3. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra+I%2C+Chapters+1%E2%80%933&rft.pub=Springer&rft.date=1989&rft.aulast=Bourbaki&rft.aufirst=N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohn2003" class="citation cs2"><a href="/wiki/Paul_Cohn" title="Paul Cohn">Cohn, Paul Moritz</a> (2003), Basic algebra: groups, rings, and fields, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-85233-587-8" title="Special:BookSources/978-1-85233-587-8"><bdi>978-1-85233-587-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+algebra%3A+groups%2C+rings%2C+and+fields&rft.pub=Springer&rft.date=2003&rft.isbn=978-1-85233-587-8&rft.aulast=Cohn&rft.aufirst=Paul+Moritz&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEisenbud1995" class="citation book cs1"><a href="/wiki/David_Eisenbud" title="David Eisenbud">Eisenbud, David</a> (1995). Commutative algebra with a view toward algebraic geometry. Graduate Texts in Mathematics. Vol. 150. Springer. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1322960">1322960</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Commutative+algebra+with+a+view+toward+algebraic+geometry&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=1995&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1322960%23id-name%3DMR&rft.aulast=Eisenbud&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGallian2006" class="citation book cs1">Gallian, Joseph A. (2006). Contemporary Abstract Algebra, Sixth Edition. Houghton Mifflin. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780618514717" title="Special:BookSources/9780618514717"><bdi>9780618514717</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Contemporary+Abstract+Algebra%2C+Sixth+Edition.&rft.pub=Houghton+Mifflin&rft.date=2006&rft.isbn=9780618514717&rft.aulast=Gallian&rft.aufirst=Joseph+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGardnerWiegandt2003" class="citation book cs1">Gardner, J.W.; Wiegandt, R. (2003). Radical Theory of Rings. Chapman & Hall/CRC Pure and Applied Mathematics. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0824750330" title="Special:BookSources/0824750330"><bdi>0824750330</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Radical+Theory+of+Rings&rft.pub=Chapman+%26+Hall%2FCRC+Pure+and+Applied+Mathematics&rft.date=2003&rft.isbn=0824750330&rft.aulast=Gardner&rft.aufirst=J.W.&rft.au=Wiegandt%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerstein1994" class="citation book cs1"><a href="/wiki/Israel_Nathan_Herstein" title="Israel Nathan Herstein">Herstein, I. N.</a> (1994) [reprint of the 1968 original]. Noncommutative rings. Carus Mathematical Monographs. Vol. 15. With an afterword by Lance W. Small. Mathematical Association of America. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-88385-015-X" title="Special:BookSources/0-88385-015-X"><bdi>0-88385-015-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Noncommutative+rings&rft.series=Carus+Mathematical+Monographs&rft.pub=Mathematical+Association+of+America&rft.date=1994&rft.isbn=0-88385-015-X&rft.aulast=Herstein&rft.aufirst=I.+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHungerford1997" class="citation book cs1">Hungerford, Thomas W. (1997). Abstract Algebra: an Introduction, Second Edition. Brooks/Cole. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780030105593" title="Special:BookSources/9780030105593"><bdi>9780030105593</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abstract+Algebra%3A+an+Introduction%2C+Second+Edition.&rft.pub=Brooks%2FCole&rft.date=1997&rft.isbn=9780030105593&rft.aulast=Hungerford&rft.aufirst=Thomas+W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacobson1964" class="citation journal cs1"><a href="/wiki/Nathan_Jacobson" title="Nathan Jacobson">Jacobson, Nathan</a> (1964). "Structure of rings". American Mathematical Society Colloquium Publications. 37 (Revised ed.).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Society+Colloquium+Publications&rft.atitle=Structure+of+rings&rft.volume=37&rft.date=1964&rft.aulast=Jacobson&rft.aufirst=Nathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacobson1943" class="citation journal cs1"><a href="/wiki/Nathan_Jacobson" title="Nathan Jacobson">Jacobson, Nathan</a> (1943). "The Theory of Rings". American Mathematical Society Mathematical Surveys. I.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Society+Mathematical+Surveys&rft.atitle=The+Theory+of+Rings&rft.volume=I&rft.date=1943&rft.aulast=Jacobson&rft.aufirst=Nathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacobson2009" class="citation book cs1"><a href="/wiki/Nathan_Jacobson" title="Nathan Jacobson">Jacobson, Nathan</a> (2009). Basic algebra. Vol. 1 (2nd ed.). Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-47189-1" title="Special:BookSources/978-0-486-47189-1"><bdi>978-0-486-47189-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Basic+algebra&rft.edition=2nd&rft.pub=Dover&rft.date=2009&rft.isbn=978-0-486-47189-1&rft.aulast=Jacobson&rft.aufirst=Nathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKaplansky1974" class="citation cs2"><a href="/wiki/Irving_Kaplansky" title="Irving Kaplansky">Kaplansky, Irving</a> (1974), <a rel="nofollow" class="external text" href="https://archive.org/details/commutativerings00irvi">Commutative rings</a> (Revised ed.), <a href="/wiki/University_of_Chicago_Press" title="University of Chicago Press">University of Chicago Press</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-226-42454-5" title="Special:BookSources/0-226-42454-5"><bdi>0-226-42454-5</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0345945">0345945</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Commutative+rings&rft.edition=Revised&rft.pub=University+of+Chicago+Press&rft.date=1974&rft.isbn=0-226-42454-5&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0345945%23id-name%3DMR&rft.aulast=Kaplansky&rft.aufirst=Irving&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcommutativerings00irvi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLam1999" class="citation book cs1"><a href="/wiki/Tsit_Yuen_Lam" title="Tsit Yuen Lam">Lam, Tsit Yuen</a> (1999). 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Algebra. AMS Chelsea.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.pub=AMS+Chelsea&rft.date=1967&rft.aulast=Mac+Lane&rft.aufirst=Saunders&rft.au=Birkhoff%2C+Garrett&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatsumura1989" class="citation book cs1">Matsumura, Hideyuki (1989). Commutative Ring Theory. Cambridge Studies in Advanced Mathematics (2nd ed.). <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-36764-6" title="Special:BookSources/978-0-521-36764-6"><bdi>978-0-521-36764-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Commutative+Ring+Theory&rft.series=Cambridge+Studies+in+Advanced+Mathematics&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=1989&rft.isbn=978-0-521-36764-6&rft.aulast=Matsumura&rft.aufirst=Hideyuki&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilne2012" class="citation web cs1">Milne, J. (2012). <a rel="nofollow" class="external text" href="http://www.jmilne.org/math/xnotes/ca.html">"A primer of commutative algebra"</a>. v2.23.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=A+primer+of+commutative+algebra&rft.date=2012&rft.aulast=Milne&rft.aufirst=J.&rft_id=http%3A%2F%2Fwww.jmilne.org%2Fmath%2Fxnotes%2Fca.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRotman1998" class="citation cs2">Rotman, Joseph (1998), Galois Theory (2nd ed.), Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-98541-7" title="Special:BookSources/0-387-98541-7"><bdi>0-387-98541-7</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Galois+Theory&rft.edition=2nd&rft.pub=Springer&rft.date=1998&rft.isbn=0-387-98541-7&rft.aulast=Rotman&rft.aufirst=Joseph&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_der_Waerden1930" class="citation cs2"><a href="/wiki/Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">van der Waerden, Bartel Leendert</a> (1930), <a href="/wiki/Moderne_Algebra" title="Moderne Algebra">Moderne Algebra. Teil I</a>, Die Grundlehren der mathematischen Wissenschaften, vol. 33, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-56799-8" title="Special:BookSources/978-3-540-56799-8"><bdi>978-3-540-56799-8</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0009016">0009016</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Moderne+Algebra.+Teil+I&rft.series=Die+Grundlehren+der+mathematischen+Wissenschaften&rft.pub=Springer&rft.date=1930&rft.isbn=978-3-540-56799-8&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0009016%23id-name%3DMR&rft.aulast=van+der+Waerden&rft.aufirst=Bartel+Leendert&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWarner1965" class="citation book cs1">Warner, Seth (1965). Modern Algebra. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486663418" title="Special:BookSources/9780486663418"><bdi>9780486663418</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Modern+Algebra&rft.pub=Dover&rft.date=1965&rft.isbn=9780486663418&rft.aulast=Warner&rft.aufirst=Seth&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilder1965" class="citation book cs1">Wilder, Raymond Louis (1965). Introduction to Foundations of Mathematics. Wiley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Foundations+of+Mathematics&rft.pub=Wiley&rft.date=1965&rft.aulast=Wilder&rft.aufirst=Raymond+Louis&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZariskiSamuel1958" class="citation book cs1">Zariski, Oscar; Samuel, Pierre (1958). Commutative Algebra. Vol. 1. 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(1947). "Anneaux finis; systèmes hypercomplexes de rang trois sur un corps commutatif". Ann. Soc. Sci. Bruxelles. I (61): 222–227.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ann.+Soc.+Sci.+Bruxelles&rft.atitle=Anneaux+finis%3B+syst%C3%A8mes+hypercomplexes+de+rang+trois+sur+un+corps+commutatif&rft.volume=I&rft.issue=61&rft.pages=222-227&rft.date=1947&rft.aulast=Ballieu&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBerrickKeating2000" class="citation book cs1">Berrick, A. J.; Keating, M. E. (2000). An Introduction to Rings and Modules with K-Theory in View. Cambridge University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Rings+and+Modules+with+K-Theory+in+View&rft.pub=Cambridge+University+Press&rft.date=2000&rft.aulast=Berrick&rft.aufirst=A.+J.&rft.au=Keating%2C+M.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohn1995" class="citation cs2"><a href="/wiki/Paul_Cohn" title="Paul Cohn">Cohn, Paul Moritz</a> (1995), <a rel="nofollow" class="external text" href="https://archive.org/details/skewfieldstheory0000cohn">Skew Fields: Theory of General Division Rings</a>, Encyclopedia of Mathematics and its Applications, vol. 57, Cambridge University Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521432177" title="Special:BookSources/9780521432177"><bdi>9780521432177</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Skew+Fields%3A+Theory+of+General+Division+Rings&rft.series=Encyclopedia+of+Mathematics+and+its+Applications&rft.pub=Cambridge+University+Press&rft.date=1995&rft.isbn=9780521432177&rft.aulast=Cohn&rft.aufirst=Paul+Moritz&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fskewfieldstheory0000cohn&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGilmerMott1973" class="citation journal cs1">Gilmer, R.; Mott, J. 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Algebra: A Graduate Course. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">AMS</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-4799-2" title="Special:BookSources/978-0-8218-4799-2"><bdi>978-0-8218-4799-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra%3A+A+Graduate+Course&rft.pub=AMS&rft.date=1994&rft.isbn=978-0-8218-4799-2&rft.aulast=Isaacs&rft.aufirst=I.+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJacobson1945" class="citation cs2"><a href="/wiki/Nathan_Jacobson" title="Nathan Jacobson">Jacobson, Nathan</a> (1945), "Structure theory of algebraic algebras of bounded degree", <a href="/wiki/Annals_of_Mathematics" title="Annals of Mathematics">Annals of Mathematics</a>, 46 (4), Annals of Mathematics: 695–707, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1969205">10.2307/1969205</a>, <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0003-486X">0003-486X</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1969205">1969205</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Mathematics&rft.atitle=Structure+theory+of+algebraic+algebras+of+bounded+degree&rft.volume=46&rft.issue=4&rft.pages=695-707&rft.date=1945&rft.issn=0003-486X&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F1969205%23id-name%3DJSTOR&rft_id=info%3Adoi%2F10.2307%2F1969205&rft.aulast=Jacobson&rft.aufirst=Nathan&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKnuth1998" class="citation book cs1 cs1-prop-long-vol"><a href="/wiki/Donald_Knuth" title="Donald Knuth">Knuth, D. E.</a> (1998). The Art of Computer Programming. Vol. 2: Seminumerical Algorithms (3rd ed.). Addison–Wesley.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Art+of+Computer+Programming&rft.edition=3rd&rft.pub=Addison%E2%80%93Wesley&rft.date=1998&rft.aulast=Knuth&rft.aufirst=D.+E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKornKorn2000" class="citation book cs1">Korn, G. A.; <a href="/wiki/Theresa_M._Korn" title="Theresa M. Korn">Korn, T. M.</a> (2000). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=xUQc0RZhQnAC&q=ring">Mathematical Handbook for Scientists and Engineers</a>. Dover. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780486411477" title="Special:BookSources/9780486411477"><bdi>9780486411477</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematical+Handbook+for+Scientists+and+Engineers&rft.pub=Dover&rft.date=2000&rft.isbn=9780486411477&rft.aulast=Korn&rft.aufirst=G.+A.&rft.au=Korn%2C+T.+M.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DxUQc0RZhQnAC%26q%3Dring&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilneCFT" class="citation web cs1">Milne, J. <a rel="nofollow" class="external text" href="http://www.jmilne.org/math/CourseNotes/cft.html">"Class field theory"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Class+field+theory&rft.aulast=Milne&rft.aufirst=J.&rft_id=http%3A%2F%2Fwww.jmilne.org%2Fmath%2FCourseNotes%2Fcft.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNagata1962" class="citation cs2"><a href="/wiki/Masayoshi_Nagata" title="Masayoshi Nagata">Nagata, Masayoshi</a> (1962) [1975 reprint], Local rings, Interscience Tracts in Pure and Applied Mathematics, vol. 13, Interscience Publishers, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-88275-228-0" title="Special:BookSources/978-0-88275-228-0"><bdi>978-0-88275-228-0</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0155856">0155856</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Local+rings&rft.series=Interscience+Tracts+in+Pure+and+Applied+Mathematics&rft.pub=Interscience+Publishers&rft.date=1962&rft.isbn=978-0-88275-228-0&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0155856%23id-name%3DMR&rft.aulast=Nagata&rft.aufirst=Masayoshi&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPierce1982" class="citation book cs1">Pierce, Richard S. (1982). <a rel="nofollow" class="external text" href="https://archive.org/details/associativealgeb00pier_0">Associative algebras</a>. Graduate Texts in Mathematics. Vol. 88. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90693-2" title="Special:BookSources/0-387-90693-2"><bdi>0-387-90693-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Associative+algebras&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=1982&rft.isbn=0-387-90693-2&rft.aulast=Pierce&rft.aufirst=Richard+S.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fassociativealgeb00pier_0&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoonen2019" class="citation cs2"><a href="/wiki/Bjorn_Poonen" title="Bjorn Poonen">Poonen, Bjorn</a> (2019), <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/48666015">"Why all rings should have a 1"</a>, Mathematics Magazine, 92 (1): 58−62, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1404.0135">1404.0135</a>, <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/48666015">48666015</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematics+Magazine&rft.atitle=Why+all+rings+should+have+a+1&rft.volume=92&rft.issue=1&rft.pages=58%E2%88%9262&rft.date=2019&rft_id=info%3Aarxiv%2F1404.0135&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F48666015%23id-name%3DJSTOR&rft.aulast=Poonen&rft.aufirst=Bjorn&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F48666015&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSerre1979" class="citation cs2"><a href="/wiki/Jean-Pierre_Serre" title="Jean-Pierre Serre">Serre, Jean-Pierre</a> (1979), Local fields, Graduate Texts in Mathematics, vol. 67, Springer</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Local+fields&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=1979&rft.aulast=Serre&rft.aufirst=Jean-Pierre&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSpringer1977" class="citation cs2">Springer, Tonny A. (1977), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pTV7CwAAQBAJ&q=ring">Invariant theory</a>, Lecture Notes in Mathematics, vol. 585, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9783540373704" title="Special:BookSources/9783540373704"><bdi>9783540373704</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Invariant+theory&rft.series=Lecture+Notes+in+Mathematics&rft.pub=Springer&rft.date=1977&rft.isbn=9783540373704&rft.aulast=Springer&rft.aufirst=Tonny+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DpTV7CwAAQBAJ%26q%3Dring&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWeibel2013" class="citation cs2">Weibel, Charles A. (2013), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Ja8xAAAAQBAJ">The K-book: An Introduction to Algebraic K-theory</a>, Graduate Studies in Mathermatics, vol. 145, American Mathematical Society, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780821891322" title="Special:BookSources/9780821891322"><bdi>9780821891322</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+K-book%3A+An+Introduction+to+Algebraic+K-theory&rft.series=Graduate+Studies+in+Mathermatics&rft.pub=American+Mathematical+Society&rft.date=2013&rft.isbn=9780821891322&rft.aulast=Weibel&rft.aufirst=Charles+A.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DJa8xAAAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"> (also <a rel="nofollow" class="external text" href="https://sites.math.rutgers.edu/~weibel/Kbook/Kbook.I.pdf">online</a>)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZariskiSamuel1975" class="citation book cs1 cs1-prop-long-vol"><a href="/wiki/Oscar_Zariski" title="Oscar Zariski">Zariski, Oscar</a>; <a href="/wiki/Pierre_Samuel" title="Pierre Samuel">Samuel, Pierre</a> (1975). Commutative algebra. Graduate Texts in Mathematics. Vol. 28–29. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-90089-6" title="Special:BookSources/0-387-90089-6"><bdi>0-387-90089-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Commutative+algebra&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer&rft.date=1975&rft.isbn=0-387-90089-6&rft.aulast=Zariski&rft.aufirst=Oscar&rft.au=Samuel%2C+Pierre&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Primary_sources">Primary sources</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=62" title="Edit section: Primary sources">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFraenkel1915" class="citation journal cs1"><a href="/wiki/Abraham_Fraenkel" title="Abraham Fraenkel">Fraenkel, A.</a> (1915). "Über die Teiler der Null und die Zerlegung von Ringen". J. Reine Angew. Math. 1915 (145): 139–176. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2Fcrll.1915.145.139">10.1515/crll.1915.145.139</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118962421">118962421</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Reine+Angew.+Math.&rft.atitle=%C3%9Cber+die+Teiler+der+Null+und+die+Zerlegung+von+Ringen&rft.volume=1915&rft.issue=145&rft.pages=139-176&rft.date=1915&rft_id=info%3Adoi%2F10.1515%2Fcrll.1915.145.139&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118962421%23id-name%3DS2CID&rft.aulast=Fraenkel&rft.aufirst=A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHilbert1897" class="citation journal cs1"><a href="/wiki/David_Hilbert" title="David Hilbert">Hilbert, David</a> (1897). "Die Theorie der algebraischen Zahlkörper". Jahresbericht der Deutschen Mathematiker-Vereinigung. 4.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Jahresbericht+der+Deutschen+Mathematiker-Vereinigung&rft.atitle=Die+Theorie+der+algebraischen+Zahlk%C3%B6rper&rft.volume=4&rft.date=1897&rft.aulast=Hilbert&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNoether1921" class="citation journal cs1"><a href="/wiki/Emmy_Noether" title="Emmy Noether">Noether, Emmy</a> (1921). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1428306">"Idealtheorie in Ringbereichen"</a>. Math. Annalen. 83 (1–2): 24–66. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fbf01464225">10.1007/bf01464225</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121594471">121594471</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Math.+Annalen&rft.atitle=Idealtheorie+in+Ringbereichen&rft.volume=83&rft.issue=1%E2%80%932&rft.pages=24-66&rft.date=1921&rft_id=info%3Adoi%2F10.1007%2Fbf01464225&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121594471%23id-name%3DS2CID&rft.aulast=Noether&rft.aufirst=Emmy&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1428306&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading3"><h3 id="Historical_references">Historical references</h3>[<a href="/w/index.php?title=Ring_(mathematics)&action=edit&section=63" title="Edit section: Historical references">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li>Bronshtein, I. N. and Semendyayev, K. A. (2004) <a href="/wiki/Bronshtein_and_Semendyayev" title="Bronshtein and Semendyayev">Handbook of Mathematics</a>, 4th ed. New York: Springer-Verlag <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-43491-7" title="Special:BookSources/3-540-43491-7">3-540-43491-7</a>.</li> <li><a rel="nofollow" class="external text" href="http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Ring_theory.html">History of ring theory at the MacTutor Archive</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBirkhoffMac_Lane1996" class="citation cs2"><a href="/wiki/Garrett_Birkhoff" title="Garrett Birkhoff">Birkhoff, Garrett</a>; <a href="/wiki/Saunders_Mac_Lane" title="Saunders Mac Lane">Mac Lane, Saunders</a> (1996), A Survey of Modern Algebra (5th ed.), New York: Macmillan</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+Survey+of+Modern+Algebra&rft.place=New+York&rft.edition=5th&rft.pub=Macmillan&rft.date=1996&rft.aulast=Birkhoff&rft.aufirst=Garrett&rft.au=Mac+Lane%2C+Saunders&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li>Faith, Carl (1999) Rings and things and a fine array of twentieth century associative algebra. Mathematical Surveys and Monographs, 65. <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8218-0993-8" title="Special:BookSources/0-8218-0993-8">0-8218-0993-8</a>.</li> <li>Itô, K. editor (1986) "Rings." §368 in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: <a href="/wiki/MIT_Press" title="MIT Press">MIT Press</a>.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleiner1996" class="citation journal cs1"><a href="/wiki/Israel_Kleiner_(mathematician)" title="Israel Kleiner (mathematician)">Kleiner, Israel</a> (1996). "The Genesis of the Abstract Ring Concept". <a href="/wiki/American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">American Mathematical Monthly</a>. 103 (5): 417–424. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2974935">10.2307/2974935</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2974935">2974935</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.atitle=The+Genesis+of+the+Abstract+Ring+Concept&rft.volume=103&rft.issue=5&rft.pages=417-424&rft.date=1996&rft_id=info%3Adoi%2F10.2307%2F2974935&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2974935%23id-name%3DJSTOR&rft.aulast=Kleiner&rft.aufirst=Israel&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKleiner1998" class="citation journal cs1"><a href="/wiki/Israel_Kleiner_(mathematician)" title="Israel Kleiner (mathematician)">Kleiner, Israel</a> (February 1998). <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs000170050029">"From Numbers to Rings: The Early History of Ring Theory"</a>. <a href="/wiki/Elemente_der_Mathematik" title="Elemente der Mathematik">Elemente der Mathematik</a>. 53 (1): 18–35. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs000170050029">10.1007/s000170050029</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Elemente+der+Mathematik&rft.atitle=From+Numbers+to+Rings%3A+The+Early+History+of+Ring+Theory&rft.volume=53&rft.issue=1&rft.pages=18-35&rft.date=1998-02&rft_id=info%3Adoi%2F10.1007%2Fs000170050029&rft.aulast=Kleiner&rft.aufirst=Israel&rft_id=https%3A%2F%2Fdoi.org%2F10.1007%252Fs000170050029&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_der_Waerden1985" class="citation cs2"><a href="/wiki/Bartel_Leendert_van_der_Waerden" title="Bartel Leendert van der Waerden">van der Waerden, B. L.</a> (1985), A History of Algebra, Springer-Verlag</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+History+of+Algebra&rft.pub=Springer-Verlag&rft.date=1985&rft.aulast=van+der+Waerden&rft.aufirst=B.+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARing+%28mathematics%29" class="Z3988"></li></ul> </div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output 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Ring (mathematics) - Wikipedia