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concepts</span> </div> </a> <button aria-controls="toc-Basic_concepts-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Basic concepts subsection</span> </button> <ul id="toc-Basic_concepts-sublist" class="vector-toc-list"> <li id="toc-Algebra_homomorphisms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebra_homomorphisms"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Algebra homomorphisms</span> </div> </a> <ul id="toc-Algebra_homomorphisms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Subalgebras_and_ideals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Subalgebras_and_ideals"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Subalgebras and ideals</span> </div> </a> <ul id="toc-Subalgebras_and_ideals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extension_of_scalars" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Extension_of_scalars"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Extension of scalars</span> </div> </a> <ul id="toc-Extension_of_scalars-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Kinds_of_algebras_and_examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Kinds_of_algebras_and_examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Kinds of algebras and examples</span> </div> </a> <button aria-controls="toc-Kinds_of_algebras_and_examples-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Kinds of algebras and examples subsection</span> </button> <ul id="toc-Kinds_of_algebras_and_examples-sublist" class="vector-toc-list"> <li id="toc-Unital_algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Unital_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Unital algebra</span> </div> </a> <ul id="toc-Unital_algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Zero_algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Zero_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Zero algebra</span> </div> </a> <ul id="toc-Zero_algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Associative_algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Associative_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Associative algebra</span> </div> </a> <ul id="toc-Associative_algebra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Non-associative_algebra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Non-associative_algebra"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Non-associative algebra</span> </div> </a> <ul id="toc-Non-associative_algebra-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Algebras_and_rings" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Algebras_and_rings"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Algebras and rings</span> </div> </a> <ul id="toc-Algebras_and_rings-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Structure_coefficients" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Structure_coefficients"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Structure coefficients</span> </div> </a> <ul id="toc-Structure_coefficients-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Classification_of_low-dimensional_unital_associative_algebras_over_the_complex_numbers" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Classification_of_low-dimensional_unital_associative_algebras_over_the_complex_numbers"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Classification of low-dimensional unital associative algebras over the complex numbers</span> </div> </a> <ul id="toc-Classification_of_low-dimensional_unital_associative_algebras_over_the_complex_numbers-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalization:_algebra_over_a_ring" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalization:_algebra_over_a_ring"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Generalization: algebra over a ring</span> </div> </a> <button aria-controls="toc-Generalization:_algebra_over_a_ring-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Generalization: algebra over a ring subsection</span> </button> <ul id="toc-Generalization:_algebra_over_a_ring-sublist" class="vector-toc-list"> <li id="toc-Associative_algebras_over_rings" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Associative_algebras_over_rings"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Associative algebras over rings</span> </div> </a> <ul id="toc-Associative_algebras_over_rings-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </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 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </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 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class="firstHeading mw-first-heading"><span class="mw-page-title-main">Algebra over a field</span></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. 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href="https://bg.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_%D0%BD%D0%B0%D0%B4_%D0%BF%D0%BE%D0%BB%D0%B5" title="Алгебра над поле – Bulgarian" lang="bg" hreflang="bg" data-title="Алгебра над поле" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%80lgebra_sobre_un_cos" title="Àlgebra sobre un cos – Catalan" lang="ca" hreflang="ca" data-title="Àlgebra sobre un cos" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_(%D1%83%D0%B9_%C3%A7%D0%B8%D0%B9%C4%95%D0%BD)" title="Алгебра (уй çийĕн) – Chuvash" lang="cv" hreflang="cv" data-title="Алгебра (уй çийĕн)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Algebra_(struktura)" title="Algebra (struktura) – Czech" lang="cs" hreflang="cs" data-title="Algebra (struktura)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Algebra_%C3%BCber_einem_K%C3%B6rper" title="Algebra über einem Körper – German" lang="de" hreflang="de" data-title="Algebra über einem Körper" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%81lgebra_sobre_un_cuerpo" title="Álgebra sobre un cuerpo – Spanish" lang="es" hreflang="es" data-title="Álgebra sobre un cuerpo" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Al%C4%9Debro" title="Alĝebro – Esperanto" lang="eo" hreflang="eo" data-title="Alĝebro" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AC%D8%A8%D8%B1_%D8%B1%D9%88%DB%8C_%DB%8C%DA%A9_%D9%85%DB%8C%D8%AF%D8%A7%D9%86" title="جبر روی یک میدان – Persian" lang="fa" hreflang="fa" data-title="جبر روی یک میدان" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Alg%C3%A8bre_sur_un_corps" title="Algèbre sur un corps – French" lang="fr" hreflang="fr" data-title="Algèbre sur un corps" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/%C3%81lxebra_sobre_un_corpo" title="Álxebra sobre un corpo – Galician" lang="gl" hreflang="gl" data-title="Álxebra sobre un corpo" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Aljabar_atas_medan" title="Aljabar atas medan – Indonesian" lang="id" hreflang="id" data-title="Aljabar atas medan" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Algebra_super_un_corpore" title="Algebra super un corpore – Interlingua" lang="ia" hreflang="ia" data-title="Algebra super un corpore" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Algebra_su_campo" title="Algebra su campo – Italian" lang="it" hreflang="it" data-title="Algebra su campo" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Algebra_(structuur)" title="Algebra (structuur) – Dutch" lang="nl" hreflang="nl" data-title="Algebra (structuur)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E4%BD%93%E4%B8%8A%E3%81%AE%E5%A4%9A%E5%85%83%E7%92%B0" title="体上の多元環 – Japanese" lang="ja" hreflang="ja" data-title="体上の多元環" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Algebra_over_ein_kropp" title="Algebra over ein kropp – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Algebra over ein kropp" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Algebra_nad_cia%C5%82em" title="Algebra nad ciałem – Polish" lang="pl" hreflang="pl" data-title="Algebra nad ciałem" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%81lgebra_sobre_um_corpo" title="Álgebra sobre um corpo – Portuguese" lang="pt" hreflang="pt" data-title="Álgebra sobre um corpo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Algebr%C4%83_peste_un_corp" title="Algebră peste un corp – Romanian" lang="ro" hreflang="ro" data-title="Algebră peste un corp" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_%D0%BD%D0%B0%D0%B4_%D0%BF%D0%BE%D0%BB%D0%B5%D0%BC" title="Алгебра над полем – Russian" lang="ru" hreflang="ru" data-title="Алгебра над полем" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Algebra_%C3%B6ver_en_kropp" title="Algebra över en kropp – Swedish" lang="sv" hreflang="sv" data-title="Algebra över en kropp" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%90%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0_%D0%BD%D0%B0%D0%B4_%D0%BF%D0%BE%D0%BB%D0%B5%D0%BC" title="Алгебра над полем – Ukrainian" lang="uk" hreflang="uk" data-title="Алгебра над полем" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/%C4%90%E1%BA%A1i_s%E1%BB%91_tr%C3%AAn_m%E1%BB%99t_tr%C6%B0%E1%BB%9Dng" title="Đại số trên một trường – Vietnamese" lang="vi" hreflang="vi" data-title="Đại số trên một trường" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E4%BB%A3%E6%95%B8_(%E4%BB%A3%E6%95%B8)" title="代數 (代數) – Literary Chinese" lang="lzh" hreflang="lzh" data-title="代數 (代數)" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a 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.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: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> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, an <b>algebra over a field</b> (often simply called an <b>algebra</b>) is a <a href="/wiki/Vector_space" title="Vector space">vector space</a> equipped with a <a href="/wiki/Bilinear_map" title="Bilinear map">bilinear</a> <a href="/wiki/Product_(mathematics)" title="Product (mathematics)">product</a>. Thus, an algebra is an <a href="/wiki/Algebraic_structure" title="Algebraic structure">algebraic structure</a> consisting of a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> together with operations of multiplication and addition and <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a> by elements of a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> and satisfying the axioms implied by "vector space" and "bilinear".<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The multiplication operation in an algebra may or may not be <a href="/wiki/Associative" class="mw-redirect" title="Associative">associative</a>, leading to the notions of <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebras</a> and <a href="/wiki/Non-associative_algebra" title="Non-associative algebra">non-associative algebras</a>. Given an integer <i>n</i>, the <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> of <a href="/wiki/Real_matrix" class="mw-redirect" title="Real matrix">real</a> <a href="/wiki/Square_matrix" title="Square matrix">square matrices</a> of order <i>n</i> is an example of an associative algebra over the field of <a href="/wiki/Real_number" title="Real number">real numbers</a> under <a href="/wiki/Matrix_addition" title="Matrix addition">matrix addition</a> and <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a> since matrix multiplication is associative. Three-dimensional <a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> with multiplication given by the <a href="/wiki/Vector_cross_product" class="mw-redirect" title="Vector cross product">vector cross product</a> is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the <a href="/wiki/Jacobi_identity" title="Jacobi identity">Jacobi identity</a> instead. </p><p>An algebra is <b>unital</b> or <b>unitary</b> if it has an <a href="/wiki/Identity_element" title="Identity element">identity element</a> with respect to the multiplication. The ring of real square matrices of order <i>n</i> forms a unital algebra since the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> of order <i>n</i> is the identity element with respect to matrix multiplication. It is an example of a unital associative algebra, a <a href="/wiki/Unital_ring" class="mw-redirect" title="Unital ring">(unital) ring</a> that is also a vector space. </p><p>Many authors use the term <i>algebra</i> to mean <i>associative algebra</i>, or <i>unital associative algebra</i>, or in some subjects such as <a href="/wiki/Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>, <i>unital associative commutative algebra</i>. </p><p>Replacing the field of scalars by a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> leads to the more general notion of an <a href="#Generalization:_algebra_over_a_ring">algebra over a ring</a>. Algebras are not to be confused with vector spaces equipped with a <a href="/wiki/Bilinear_form" title="Bilinear form">bilinear form</a>, like <a href="/wiki/Inner_product_space" title="Inner product space">inner product spaces</a>, as, for such a space, the result of a product is not in the space, but rather in the field of coefficients. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition_and_motivation">Definition and motivation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=1" title="Edit section: Definition and motivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Motivating_examples">Motivating examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=2" title="Edit section: Motivating examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="wikitable"> <caption> </caption> <tbody><tr> <th>Algebra </th> <th>vector space </th> <th>bilinear operator </th> <th><a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associativity</a> </th> <th><a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutativity</a> </th></tr> <tr> <td><a href="/wiki/Complex_number" title="Complex number">complex numbers</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"></span> </td> <td><a href="/wiki/Complex_number#Multiplication" title="Complex number">product of complex numbers</a><br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(a+ib\right)\cdot \left(c+id\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mrow> <mo>)</mo> </mrow> <mo>⋅</mo> <mrow> <mo>(</mo> <mrow> <mi>c</mi> <mo>+</mo> <mi>i</mi> <mi>d</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(a+ib\right)\cdot \left(c+id\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/074ffa6a1894fd2c083a1bdf210d11078d17282c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.033ex; height:2.843ex;" alt="{\displaystyle \left(a+ib\right)\cdot \left(c+id\right)}"></span> </td> <td>Yes </td> <td>Yes </td></tr> <tr> <td><a href="/wiki/Cross_product" title="Cross product">cross product</a> of 3D vectors </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"></span> </td> <td><a href="/wiki/Cross_product" title="Cross product">cross product</a> <br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {a}}\times {\vec {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>b</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {a}}\times {\vec {b}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68affde71ee9058a22787425a052cf5a13f33f4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.164ex; height:2.843ex;" alt="{\displaystyle {\vec {a}}\times {\vec {b}}}"></span> </td> <td>No </td> <td>No (<a href="/wiki/Anticommutative_property" title="Anticommutative property">anticommutative</a>) </td></tr> <tr> <td><a href="/wiki/Quaternion" title="Quaternion">quaternions</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{4}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{4}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4abb9b9dab94f7b25a4210364f0f9032704bfb9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{4}}"></span> </td> <td><a href="/wiki/Quaternion#Hamilton_product" title="Quaternion">Hamilton product</a> <br /><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a+{\vec {v}})(b+{\vec {w}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>v</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>b</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>w</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a+{\vec {v}})(b+{\vec {w}})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/956c363928b803c44c2e08f1d286be53d80aab1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.366ex; height:2.843ex;" alt="{\displaystyle (a+{\vec {v}})(b+{\vec {w}})}"></span> </td> <td>Yes </td> <td>No </td></tr> <tr> <td><a href="/wiki/Polynomial" title="Polynomial">polynomials</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} [X]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} [X]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16d740527b0b7f949b4bf9c9ce004134bb490b68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.952ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} [X]}"></span> </td> <td><a href="/wiki/Polynomial_multiplication" class="mw-redirect" title="Polynomial multiplication">polynomial multiplication</a> </td> <td>Yes </td> <td>Yes </td></tr> <tr> <td><a href="/wiki/Square_matrix" title="Square matrix">square matrices</a> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n\times n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n\times n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7095e87521f2c247d021fa7101072f11beba0a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.161ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n\times n}}"></span> </td> <td><a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a> </td> <td>Yes </td> <td>No </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=3" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <span class="texhtml mvar" style="font-style:italic;">K</span> be a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, and let <span class="texhtml mvar" style="font-style:italic;">A</span> be a <a href="/wiki/Vector_space" title="Vector space">vector space</a> over <span class="texhtml mvar" style="font-style:italic;">K</span> equipped with an additional <a href="/wiki/Binary_operation" title="Binary operation">binary operation</a> from <span class="texhtml"><i>A</i> × <i>A</i></span> to <span class="texhtml mvar" style="font-style:italic;">A</span>, denoted here by <span class="texhtml">·</span> (that is, if <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span> are any two elements of <span class="texhtml mvar" style="font-style:italic;">A</span>, then <span class="texhtml"><i>x</i> · <i>y</i></span> is an element of <span class="texhtml mvar" style="font-style:italic;">A</span> that is called the <i>product</i> of <span class="texhtml mvar" style="font-style:italic;">x</span> and <span class="texhtml mvar" style="font-style:italic;">y</span>). Then <span class="texhtml mvar" style="font-style:italic;">A</span> is an <i>algebra</i> over <span class="texhtml mvar" style="font-style:italic;">K</span> if the following identities hold for all elements <span class="texhtml"><i>x</i>, <i>y</i>, <i>z</i></span> in <span class="texhtml mvar" style="font-style:italic;">A</span> , and all elements (often called <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a>) <span class="texhtml mvar" style="font-style:italic;">a</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> in <span class="texhtml mvar" style="font-style:italic;">K</span>: </p> <ul><li>Right <a href="/wiki/Distributivity" class="mw-redirect" title="Distributivity">distributivity</a>: <span class="texhtml">(<i>x</i> + <i>y</i>) · <i>z</i> = <i>x</i> · <i>z</i> + <i>y</i> · <i>z</i></span></li> <li>Left distributivity: <span class="texhtml"><i>z</i> · (<i>x</i> + <i>y</i>) = <i>z</i> · <i>x</i> + <i>z</i> · <i>y</i></span></li> <li>Compatibility with scalars: <span class="texhtml">(<i>ax</i>) · (<i>by</i>) = (<i>ab</i>) (<i>x</i> · <i>y</i>)</span>.</li></ul> <p>These three axioms are another way of saying that the binary operation is <a href="/wiki/Bilinear_operator" class="mw-redirect" title="Bilinear operator">bilinear</a>. An algebra over <span class="texhtml mvar" style="font-style:italic;">K</span> is sometimes also called a <i><span class="texhtml mvar" style="font-style:italic;">K</span>-algebra</i>, and <span class="texhtml mvar" style="font-style:italic;">K</span> is called the <i>base field</i> of <span class="texhtml mvar" style="font-style:italic;">A</span>. The binary operation is often referred to as <i>multiplication</i> in <span class="texhtml mvar" style="font-style:italic;">A</span>. The convention adopted in this article is that multiplication of elements of an algebra is not necessarily <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associative</a>, although some authors use the term <i>algebra</i> to refer to an <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebra</a>. </p><p>When a binary operation on a vector space is <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a>, left distributivity and right distributivity are equivalent, and, in this case, only one distributivity requires a proof. In general, for non-commutative operations left distributivity and right distributivity are not equivalent, and require separate proofs. </p> <div class="mw-heading mw-heading2"><h2 id="Basic_concepts">Basic concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=4" title="Edit section: Basic concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Algebra_homomorphisms">Algebra homomorphisms</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=5" title="Edit section: Algebra homomorphisms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given <span class="texhtml"><i>K</i></span>-algebras <span class="texhtml"><i>A</i></span> and <span class="texhtml"><i>B</i></span>, a <a href="/wiki/Homomorphism" title="Homomorphism">homomorphism</a> of <span class="texhtml"><i>K</i></span>-algebras or <span class="texhtml"><i>K</i></span>-<b>algebra homomorphism</b> is a <span class="texhtml"><i>K</i></span>-<a href="/wiki/Linear_map" title="Linear map">linear map</a> <span class="texhtml"><i>f</i>: <i>A</i> → <i>B</i></span> such that <span class="texhtml"><i>f</i>(<i>xy</i>) = <i>f</i>(<i>x</i>) <i>f</i>(<i>y</i>)</span> for all <span class="texhtml"><i>x</i>, <i>y</i></span> in <span class="texhtml"><i>A</i></span>. If <span class="texhtml"><i>A</i></span> and <span class="texhtml"><i>B</i></span> are unital, then a homomorphism satisfying <span class="texhtml"><i>f</i>(1<sub><i>A</i></sub>) = 1<sub><i>B</i></sub></span> is said to be a unital homomorphism. The space of all <span class="texhtml"><i>K</i></span>-algebra homomorphisms between <span class="texhtml"><i>A</i></span> and <span class="texhtml"><i>B</i></span> is frequently written as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {Hom} _{K{\text{-alg}}}(A,B).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">H</mi> <mi mathvariant="bold">o</mi> <mi mathvariant="bold">m</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>-alg</mtext> </mrow> </mrow> </msub> <mo stretchy="false">(</mo> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {Hom} _{K{\text{-alg}}}(A,B).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bbb4aa0b73ba8132d44bafdba9cdc4b054f6c2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.994ex; height:3.009ex;" alt="{\displaystyle \mathbf {Hom} _{K{\text{-alg}}}(A,B).}"></span></dd></dl> <p>A <span class="texhtml"><i>K</i></span>-algebra <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> is a <a href="/wiki/Bijective" class="mw-redirect" title="Bijective">bijective</a> <span class="texhtml"><i>K</i></span>-algebra homomorphism. </p> <div class="mw-heading mw-heading3"><h3 id="Subalgebras_and_ideals">Subalgebras and ideals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=6" title="Edit section: Subalgebras and ideals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/Substructure_(mathematics)" title="Substructure (mathematics)">Substructure (mathematics)</a></div> <p>A <i>subalgebra</i> of an algebra over a field <i>K</i> is a <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a> that has the property that the product of any two of its elements is again in the subspace. In other words, a subalgebra of an algebra is a non-empty subset of elements that is closed under addition, multiplication, and scalar multiplication. In symbols, we say that a subset <i>L</i> of a <i>K</i>-algebra <i>A</i> is a subalgebra if for every <i>x</i>, <i>y</i> in <i>L</i> and <i>c</i> in <i>K</i>, we have that <i>x</i> · <i>y</i>, <i>x</i> + <i>y</i>, and <i>cx</i> are all in <i>L</i>. </p><p>In the above example of the complex numbers viewed as a two-dimensional algebra over the real numbers, the one-dimensional real line is a subalgebra. </p><p>A <i>left ideal</i> of a <i>K</i>-algebra is a linear subspace that has the property that any element of the subspace multiplied on the left by any element of the algebra produces an element of the subspace. In symbols, we say that a subset <i>L</i> of a <i>K</i>-algebra <i>A</i> is a left ideal if for every <i>x</i> and <i>y</i> in <i>L</i>, <i>z</i> in <i>A</i> and <i>c</i> in <i>K</i>, we have the following three statements. </p> <ol><li><i>x</i> + <i>y</i> is in <i>L</i> (<i>L</i> is closed under addition),</li> <li><i>cx</i> is in <i>L</i> (<i>L</i> is closed under scalar multiplication),</li> <li><i>z</i> · <i>x</i> is in <i>L</i> (<i>L</i> is closed under left multiplication by arbitrary elements).</li></ol> <p>If (3) were replaced with <i>x</i> · <i>z</i> is in <i>L</i>, then this would define a <i>right ideal</i>. A <i>two-sided ideal</i> is a subset that is both a left and a right ideal. The term <i>ideal</i> on its own is usually taken to mean a two-sided ideal. Of course when the algebra is commutative, then all of these notions of ideal are equivalent. Conditions (1) and (2) together are equivalent to <i>L</i> being a linear subspace of <i>A</i>. It follows from condition (3) that every left or right ideal is a subalgebra. </p><p>This definition is different from the definition of an <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal of a ring</a>, in that here we require the condition (2). Of course if the algebra is unital, then condition (3) implies condition (2). </p> <div class="mw-heading mw-heading3"><h3 id="Extension_of_scalars">Extension of scalars</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=7" title="Edit section: Extension of scalars"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Extension_of_scalars" class="mw-redirect" title="Extension of scalars">Extension of scalars</a></div> <p>If we have a <a href="/wiki/Field_extension" title="Field extension">field extension</a> <i>F</i>/<i>K</i>, which is to say a bigger field <i>F</i> that contains <i>K</i>, then there is a natural way to construct an algebra over <i>F</i> from any algebra over <i>K</i>. It is the same construction one uses to make a vector space over a bigger field, namely the tensor product <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{F}:=V\otimes _{K}F}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> <mo>:=</mo> <mi>V</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 V_{F}:=V\otimes _{K}F}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/335f8f1ccf450231a41988d52133d689c6a17b2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.625ex; height:2.509ex;" alt="{\displaystyle V_{F}:=V\otimes _{K}F}"></span>. So if <i>A</i> is an algebra over <i>K</i>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{F}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>F</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{F}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f64867539242b9693f2154d734886d054b39f0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.206ex; height:2.509ex;" alt="{\displaystyle A_{F}}"></span> is an algebra over <i>F</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Kinds_of_algebras_and_examples">Kinds of algebras and examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=8" title="Edit section: Kinds of algebras and examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Algebras over fields come in many different types. These types are specified by insisting on some further axioms, such as <a href="/wiki/Commutativity" class="mw-redirect" title="Commutativity">commutativity</a> or <a href="/wiki/Associativity" class="mw-redirect" title="Associativity">associativity</a> of the multiplication operation, which are not required in the broad definition of an algebra. The theories corresponding to the different types of algebras are often very different. </p> <div class="mw-heading mw-heading3"><h3 id="Unital_algebra">Unital algebra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=9" title="Edit section: Unital algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An algebra is <i>unital</i> or <i>unitary</i> if it has a <a href="/wiki/Unit_(algebra)" class="mw-redirect" title="Unit (algebra)">unit</a> or identity element <i>I</i> with <i>Ix</i> = <i>x</i> = <i>xI</i> for all <i>x</i> in the algebra. </p> <div class="mw-heading mw-heading3"><h3 id="Zero_algebra">Zero algebra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=10" title="Edit section: Zero algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An algebra is called a <b>zero algebra</b> if <span class="nowrap"><i>uv</i> = 0</span> for all <i>u</i>, <i>v</i> in the algebra,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> not to be confused with the algebra with one element. It is inherently non-unital (except in the case of only one element), associative and commutative. </p><p>One may define a <b>unital zero algebra</b> by taking the <a href="/wiki/Direct_sum_of_modules" title="Direct sum of modules">direct sum of modules</a> of a field (or more generally a ring) <i>K</i> and a <i>K</i>-vector space (or module) <i>V</i>, and defining the product of every pair of elements of <i>V</i> to be zero. That is, if <span class="nowrap"><i>λ</i>, <i>μ</i> ∈ <i>K</i></span> and <span class="nowrap"><i>u</i>, <i>v</i> ∈ <i>V</i></span>, then <span class="nowrap">(<i>λ</i> + <i>u</i>) (<i>μ</i> + <i>v</i>) = <i>λμ</i> + (<i>λv</i> + <i>μu</i>)</span>. If <span class="nowrap"><i>e</i><sub>1</sub>, ... <i>e</i><sub><i>d</i></sub></span> is a basis of <i>V</i>, the unital zero algebra is the quotient of the polynomial ring <span class="nowrap"><i>K</i>[<i>E</i><sub>1</sub>, ..., <i>E</i><sub><i>n</i></sub>]</span> by the <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> generated by the <i>E</i><sub><i>i</i></sub><i>E</i><sub><i>j</i></sub> for every pair <span class="nowrap">(<i>i</i>, <i>j</i>)</span>. </p><p>An example of unital zero algebra is the algebra of <a href="/wiki/Dual_number" title="Dual number">dual numbers</a>, the unital zero <b>R</b>-algebra built from a one dimensional real vector space. </p><p>These unital zero algebras may be more generally useful, as they allow to translate any general property of the algebras to properties of vector spaces or <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a>. For example, the theory of <a href="/wiki/Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner bases</a> was introduced by <a href="/wiki/Bruno_Buchberger" title="Bruno Buchberger">Bruno Buchberger</a> for <a href="/wiki/Ideal_(ring_theory)" title="Ideal (ring theory)">ideals</a> in a polynomial ring <span class="nowrap"><i>R</i> = <i>K</i>[<i>x</i><sub>1</sub>, ..., <i>x</i><sub><i>n</i></sub>]</span> over a field. The construction of the unital zero algebra over a free <i>R</i>-module allows extending this theory as a Gröbner basis theory for submodules of a free module. This extension allows, for computing a Gröbner basis of a submodule, to use, without any modification, any algorithm and any software for computing Gröbner bases of ideals. </p> <div class="mw-heading mw-heading3"><h3 id="Associative_algebra">Associative algebra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=11" title="Edit section: Associative algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Associative_algebra" title="Associative algebra">Associative algebra</a></div> <p>Examples of associative algebras include </p> <ul><li>the algebra of all <i>n</i>-by-<i>n</i> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> over a field (or commutative ring) <i>K</i>. Here the multiplication is ordinary <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>.</li> <li><a href="/wiki/Group_ring" title="Group ring">group algebras</a>, where a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> serves as a basis of the vector space and algebra multiplication extends group multiplication.</li> <li>the commutative algebra <i>K</i>[<i>x</i>] of all <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> over <i>K</i> (see <a href="/wiki/Polynomial_ring" title="Polynomial ring">polynomial ring</a>).</li> <li>algebras of <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a>, such as the <b>R</b>-algebra of all real-valued <a href="/wiki/Continuous_function" title="Continuous function">continuous</a> functions defined on the <a href="/wiki/Interval_(mathematics)" title="Interval (mathematics)">interval</a> [0,1], or the <b>C</b>-algebra of all <a href="/wiki/Holomorphic_function" title="Holomorphic function">holomorphic functions</a> defined on some fixed open set in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>. These are also commutative.</li> <li><a href="/wiki/Incidence_algebra" title="Incidence algebra">Incidence algebras</a> are built on certain <a href="/wiki/Partially_ordered_set" title="Partially ordered set">partially ordered sets</a>.</li> <li>algebras of <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operators</a>, for example on a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a>. Here the algebra multiplication is given by the <a href="/wiki/Functional_composition" class="mw-redirect" title="Functional composition">composition</a> of operators. These algebras also carry a <a href="/wiki/Topological_space" title="Topological space">topology</a>; many of them are defined on an underlying <a href="/wiki/Banach_space" title="Banach space">Banach space</a>, which turns them into <a href="/wiki/Banach_algebra" title="Banach algebra">Banach algebras</a>. If an involution is given as well, we obtain <a href="/wiki/B*-algebra" class="mw-redirect" title="B*-algebra">B*-algebras</a> and <a href="/wiki/C*-algebra" title="C*-algebra">C*-algebras</a>. These are studied in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a>.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Non-associative_algebra">Non-associative algebra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=12" title="Edit section: Non-associative algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Non-associative_algebra" title="Non-associative algebra">Non-associative algebra</a></div> <p>A <i>non-associative algebra</i><sup id="cite_ref-Schafer_3-0" class="reference"><a href="#cite_note-Schafer-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> (or <i>distributive algebra</i>) over a field <i>K</i> is a <i>K</i>-vector space <i>A</i> equipped with a <i>K</i>-<a href="/wiki/Bilinear_map" title="Bilinear map">bilinear map</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\times A\rightarrow A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>×</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\times A\rightarrow A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52b681b751812bf4885e51e18c8a6f15db5b0478" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.684ex; height:2.176ex;" alt="{\displaystyle A\times A\rightarrow A}"></span>. The usage of "non-associative" here is meant to convey that associativity is not assumed, but it does not mean it is prohibited – that is, it means "not necessarily associative". </p><p>Examples detailed in the main article include: </p> <ul><li><a href="/wiki/Euclidean_space" title="Euclidean space">Euclidean space</a> <b>R</b><sup>3</sup> with multiplication given by the <a href="/wiki/Vector_cross_product" class="mw-redirect" title="Vector cross product">vector cross product</a></li> <li><a href="/wiki/Octonion" title="Octonion">Octonions</a></li> <li><a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebras</a></li> <li><a href="/wiki/Jordan_algebra" title="Jordan algebra">Jordan algebras</a></li> <li><a href="/wiki/Alternative_algebra" title="Alternative algebra">Alternative algebras</a></li> <li><a href="/wiki/Flexible_algebra" title="Flexible algebra">Flexible algebras</a></li> <li><a href="/wiki/Power-associative_algebra" class="mw-redirect" title="Power-associative algebra">Power-associative algebras</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Algebras_and_rings">Algebras and rings</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=13" title="Edit section: Algebras and rings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The definition of an associative <i>K</i>-algebra with unit is also frequently given in an alternative way. In this case, an algebra over a field <i>K</i> is a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> <i>A</i> together with a <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta \colon K\to Z(A),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>:</mo> <mi>K</mi> <mo stretchy="false">→</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta \colon K\to Z(A),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eb4c3fe1b9f4976f793d64d94d0ddcf1a246aa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.763ex; height:2.843ex;" alt="{\displaystyle \eta \colon K\to Z(A),}"></span></dd></dl> <p>where <i>Z</i>(<i>A</i>) is the <a href="/wiki/Center_(ring_theory)" title="Center (ring theory)">center</a> of <i>A</i>. Since <i>η</i> is a ring homomorphism, then one must have either that <i>A</i> is the <a href="/wiki/Zero_ring" title="Zero ring">zero ring</a>, or that <i>η</i> is <a href="/wiki/Injective_function" title="Injective function">injective</a>. This definition is equivalent to that above, with scalar multiplication </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\times A\to A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> <mo>×</mo> <mi>A</mi> <mo stretchy="false">→</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K\times A\to A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c523bfe1944d4134d5e6e134ddf0ff9117faf79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.007ex; height:2.176ex;" alt="{\displaystyle K\times A\to A}"></span></dd></dl> <p>given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (k,a)\mapsto \eta (k)a.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo stretchy="false">↦</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (k,a)\mapsto \eta (k)a.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c36ba6340deb08edbc3fb39b75a2a2420f7d8eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.965ex; height:2.843ex;" alt="{\displaystyle (k,a)\mapsto \eta (k)a.}"></span></dd></dl> <p>Given two such associative unital <i>K</i>-algebras <i>A</i> and <i>B</i>, a unital <i>K</i>-algebra homomorphism <i>f</i>: <i>A</i> → <i>B</i> is a ring homomorphism that commutes with the scalar multiplication defined by <i>η</i>, which one may write as </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(ka)=kf(a)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(ka)=kf(a)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/395e0295fcb78f6f5fc4ca8f769e19a50d207f80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.156ex; height:2.843ex;" alt="{\displaystyle f(ka)=kf(a)}"></span></dd></dl> <p>for all <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈</mo> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3846561ca971f4b17e10153d1f996e08e5ac192" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.118ex; height:2.176ex;" alt="{\displaystyle k\in K}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a97387981adb5d65f74518e20b6785a284d7abd5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}"></span>. In other words, the following diagram commutes: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}&&K&&\\&\eta _{A}\swarrow &\,&\eta _{B}\searrow &\\A&&{\begin{matrix}f\\\longrightarrow \end{matrix}}&&B\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd /> <mtd /> <mtd> <mi>K</mi> </mtd> <mtd /> <mtd /> </mtr> <mtr> <mtd /> <mtd> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">↙</mo> </mtd> <mtd> <mspace width="thinmathspace" /> </mtd> <mtd> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mo stretchy="false">↘</mo> </mtd> <mtd /> </mtr> <mtr> <mtd> <mi>A</mi> </mtd> <mtd /> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>f</mi> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⟶</mo> </mtd> </mtr> </mtable> </mrow> </mtd> <mtd /> <mtd> <mi>B</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}&&K&&\\&\eta _{A}\swarrow &\,&\eta _{B}\searrow &\\A&&{\begin{matrix}f\\\longrightarrow \end{matrix}}&&B\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11e2a5ca464b5caac557a92327a5e8484c05b496" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:29.299ex; height:12.509ex;" alt="{\displaystyle {\begin{matrix}&&K&&\\&\eta _{A}\swarrow &\,&\eta _{B}\searrow &\\A&&{\begin{matrix}f\\\longrightarrow \end{matrix}}&&B\end{matrix}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Structure_coefficients">Structure coefficients</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=14" title="Edit section: Structure coefficients"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Structure_constants" title="Structure constants">Structure constants</a></div> <p>For algebras over a field, the bilinear multiplication from <i>A</i> × <i>A</i> to <i>A</i> is completely determined by the multiplication of <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> elements of <i>A</i>. Conversely, once a basis for <i>A</i> has been chosen, the products of basis elements can be set arbitrarily, and then extended in a unique way to a bilinear operator on <i>A</i>, i.e., so the resulting multiplication satisfies the algebra laws. </p><p>Thus, given the field <i>K</i>, any finite-dimensional algebra can be specified <a href="/wiki/Up_to" title="Up to">up to</a> <a href="/wiki/Isomorphism" title="Isomorphism">isomorphism</a> by giving its <a href="/wiki/Dimension_(linear_algebra)" class="mw-redirect" title="Dimension (linear algebra)">dimension</a> (say <i>n</i>), and specifying <i>n</i><sup>3</sup> <i>structure coefficients</i> <i>c</i><sub><i>i</i>,<i>j</i>,<i>k</i></sub>, which are <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a>. These structure coefficients determine the multiplication in <i>A</i> via the following rule: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{i}\mathbf {e} _{j}=\sum _{k=1}^{n}c_{i,j,k}\mathbf {e} _{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo>,</mo> <mi>k</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">e</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {e} _{i}\mathbf {e} _{j}=\sum _{k=1}^{n}c_{i,j,k}\mathbf {e} _{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cd419e8cacc6b9ab0175c2938c6243c9a5655d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:17.569ex; height:6.843ex;" alt="{\displaystyle \mathbf {e} _{i}\mathbf {e} _{j}=\sum _{k=1}^{n}c_{i,j,k}\mathbf {e} _{k}}"></span></dd></dl> <p>where <b>e</b><sub>1</sub>,...,<b>e</b><sub><i>n</i></sub> form a basis of <i>A</i>. </p><p>Note however that several different sets of structure coefficients can give rise to isomorphic algebras. </p><p>In <a href="/wiki/Mathematical_physics" title="Mathematical physics">mathematical physics</a>, the structure coefficients are generally written with upper and lower indices, so as to distinguish their transformation properties under coordinate transformations. Specifically, lower indices are <a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">covariant</a> indices, and transform via <a href="/wiki/Pullback_(differential_geometry)" title="Pullback (differential geometry)">pullbacks</a>, while upper indices are <a href="/wiki/Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">contravariant</a>, transforming under <a href="/wiki/Pushforward_(differential)" title="Pushforward (differential)">pushforwards</a>. Thus, the structure coefficients are often written <i>c</i><sub><i>i</i>,<i>j</i></sub><sup><i>k</i></sup>, and their defining rule is written using the <a href="/wiki/Einstein_notation" title="Einstein notation">Einstein notation</a> as </p> <dl><dd><b>e</b><sub><i>i</i></sub><b>e</b><sub><i>j</i></sub> = <i>c</i><sub><i>i</i>,<i>j</i></sub><sup><i>k</i></sup><b>e</b><sub><i>k</i></sub>.</dd></dl> <p>If you apply this to vectors written in <a href="/wiki/Index_notation" title="Index notation">index notation</a>, then this becomes </p> <dl><dd>(<b>xy</b>)<sup><i>k</i></sup> = <i>c</i><sub><i>i</i>,<i>j</i></sub><sup><i>k</i></sup><i>x</i><sup><i>i</i></sup><i>y</i><sup><i>j</i></sup>.</dd></dl> <p>If <i>K</i> is only a commutative ring and not a field, then the same process works if <i>A</i> is a <a href="/wiki/Free_module" title="Free module">free module</a> over <i>K</i>. If it isn't, then the multiplication is still completely determined by its action on a set that spans <i>A</i>; however, the structure constants can't be specified arbitrarily in this case, and knowing only the structure constants does not specify the algebra up to isomorphism. </p> <div class="mw-heading mw-heading2"><h2 id="Classification_of_low-dimensional_unital_associative_algebras_over_the_complex_numbers">Classification of low-dimensional unital associative algebras over the complex numbers</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=15" title="Edit section: Classification of low-dimensional unital associative algebras over the complex numbers"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Two-dimensional, three-dimensional and four-dimensional unital associative algebras over the field of complex numbers were completely classified up to isomorphism by <a href="/wiki/Eduard_Study" title="Eduard Study">Eduard Study</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>There exist two such two-dimensional algebras. Each algebra consists of linear combinations (with complex coefficients) of two basis elements, 1 (the identity element) and <i>a</i>. According to the definition of an identity element, </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle 1\cdot 1=1\,,\quad 1\cdot a=a\,,\quad a\cdot 1=a\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mn>1</mn> <mo>⋅</mo> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mo>⋅</mo> <mn>1</mn> <mo>=</mo> <mi>a</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle 1\cdot 1=1\,,\quad 1\cdot a=a\,,\quad a\cdot 1=a\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/303a2e8eb62745c282986dfb92af18fb94ffa843" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.585ex; height:2.509ex;" alt="{\displaystyle \textstyle 1\cdot 1=1\,,\quad 1\cdot a=a\,,\quad a\cdot 1=a\,.}"></span></dd></dl> <p>It remains to specify </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle aa=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle aa=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/096137d97e4450c56e0ef14a94baf8a3fa81dda5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.721ex; height:2.176ex;" alt="{\displaystyle \textstyle aa=1}"></span>   for the first algebra,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle aa=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle aa=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2a908796e3b3efe0875a3189975aa2f93911da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.721ex; height:2.176ex;" alt="{\displaystyle \textstyle aa=0}"></span>   for the second algebra.</dd></dl> <p>There exist five such three-dimensional algebras. Each algebra consists of linear combinations of three basis elements, 1 (the identity element), <i>a</i> and <i>b</i>. Taking into account the definition of an identity element, it is sufficient to specify </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle aa=a\,,\quad bb=b\,,\quad ab=ba=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>b</mi> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle aa=a\,,\quad bb=b\,,\quad ab=ba=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62b252e95547e5edafe98a2127c409389824d921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.18ex; height:2.509ex;" alt="{\displaystyle \textstyle aa=a\,,\quad bb=b\,,\quad ab=ba=0}"></span>   for the first algebra,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle aa=a\,,\quad bb=0\,,\quad ab=ba=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mi>a</mi> <mo>=</mo> <mi>a</mi> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>b</mi> <mi>b</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle aa=a\,,\quad bb=0\,,\quad ab=ba=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f12c6baf3854a423e99cbd673221b75d8a1e3294" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.345ex; height:2.509ex;" alt="{\displaystyle \textstyle aa=a\,,\quad bb=0\,,\quad ab=ba=0}"></span>   for the second algebra,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle aa=b\,,\quad bb=0\,,\quad ab=ba=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mi>a</mi> <mo>=</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>b</mi> <mi>b</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle aa=b\,,\quad bb=0\,,\quad ab=ba=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e24864066dbb2bddb3c282455024397851c709ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.113ex; height:2.509ex;" alt="{\displaystyle \textstyle aa=b\,,\quad bb=0\,,\quad ab=ba=0}"></span>   for the third algebra,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle aa=1\,,\quad bb=0\,,\quad ab=-ba=b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>b</mi> <mi>b</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mo>−</mo> <mi>b</mi> <mi>a</mi> <mo>=</mo> <mi>b</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle aa=1\,,\quad bb=0\,,\quad ab=-ba=b}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1242042b35d0f53f0a70a37c8c46c4cdc5e33552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:33.921ex; height:2.509ex;" alt="{\displaystyle \textstyle aa=1\,,\quad bb=0\,,\quad ab=-ba=b}"></span>   for the fourth algebra,</dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle aa=0\,,\quad bb=0\,,\quad ab=ba=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mi>a</mi> <mi>a</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>b</mi> <mi>b</mi> <mo>=</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>a</mi> <mi>b</mi> <mo>=</mo> <mi>b</mi> <mi>a</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle aa=0\,,\quad bb=0\,,\quad ab=ba=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15f040db4b1c9bf9e1028c5771f33d88a0761ae6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:32.278ex; height:2.509ex;" alt="{\displaystyle \textstyle aa=0\,,\quad bb=0\,,\quad ab=ba=0}"></span>   for the fifth algebra.</dd></dl> <p>The fourth of these algebras is non-commutative, and the others are commutative. </p> <div class="mw-heading mw-heading2"><h2 id="Generalization:_algebra_over_a_ring">Generalization: algebra over a ring</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=16" title="Edit section: Generalization: algebra over a ring"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In some areas of mathematics, such as <a href="/wiki/Commutative_algebra" title="Commutative algebra">commutative algebra</a>, it is common to consider the more general concept of an <b>algebra over a ring</b>, where a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a> <i>R</i> replaces the field <i>K</i>. The only part of the definition that changes is that <i>A</i> is assumed to be an <a href="/wiki/Module_(mathematics)" title="Module (mathematics)"><i>R</i>-module</a> (instead of a <i>K</i>-vector space). </p> <div class="mw-heading mw-heading3"><h3 id="Associative_algebras_over_rings">Associative algebras over rings</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=17" title="Edit section: Associative algebras over rings"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Associative_algebra" title="Associative algebra">Associative algebra</a></div> <p>A <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> <i>A</i> is always an associative algebra over its <a href="/wiki/Center_(ring_theory)" title="Center (ring theory)">center</a>, and over the <a href="/wiki/Integer" title="Integer">integers</a>. A classical example of an algebra over its center is the <a href="/wiki/Split-biquaternion" title="Split-biquaternion">split-biquaternion algebra</a>, which is isomorphic to <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {H} \times \mathbb {H} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">H</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {H} \times \mathbb {H} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bc7d76fd884119d7debff60d92de9fcc539a905" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.457ex; height:2.176ex;" alt="{\displaystyle \mathbb {H} \times \mathbb {H} }"></span>, the direct product of two <a href="/wiki/Quaternion" title="Quaternion">quaternion algebras</a>. The center of that ring is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \times \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> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \times \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb99f01c438a62e4ac5af8cff4eb402739ed67a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} \times \mathbb {R} }"></span>, and hence it has the structure of an algebra over its center, which is not a field. Note that the split-biquaternion algebra is also naturally an 8-dimensional <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>-algebra. </p><p>In commutative algebra, if <i>A</i> is a <a href="/wiki/Commutative_ring" title="Commutative ring">commutative ring</a>, then any unital ring homomorphism <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\to A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">→</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\to A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c29adbb31d030193e6c294c7f438b0c64753ed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.121ex; height:2.176ex;" alt="{\displaystyle R\to A}"></span> defines an <i>R</i>-module structure on <i>A</i>, and this is what is known as the <i>R</i>-algebra structure.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> So a ring comes with a natural <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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} }"></span>-module structure, since one can take the unique homomorphism <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} \to A}"> <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>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \to A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcf64d6f11aeb1d91b42b8ff852b81aa0fad6cfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.908ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} \to A}"></span>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> On the other hand, not all rings can be given the structure of an algebra over a field (for example the integers). See <i><a href="/wiki/Field_with_one_element" title="Field with one element">Field with one element</a></i> for a description of an attempt to give to every ring a structure that behaves like an algebra over a field. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=18" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Algebra_over_an_operad" class="mw-redirect" title="Algebra over an operad">Algebra over an operad</a></li> <li><a href="/wiki/Alternative_algebra" title="Alternative algebra">Alternative algebra</a></li> <li><a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a></li> <li><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebra</a></li> <li><a href="/wiki/Differential_algebra" title="Differential algebra">Differential algebra</a></li> <li><a href="/wiki/Free_algebra" title="Free algebra">Free algebra</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></li> <li><a href="/wiki/Max-plus_algebra" class="mw-redirect" title="Max-plus algebra">Max-plus algebra</a></li> <li><a href="/wiki/Mutation_(algebra)" title="Mutation (algebra)">Mutation (algebra)</a></li> <li><a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></li> <li><a href="/wiki/Zariski%27s_lemma" title="Zariski's lemma">Zariski's lemma</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=19" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">See also <a href="#CITEREFHazewinkelGubareniKirichenko2004">Hazewinkel, Gubareni & Kirichenko 2004</a>, p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=AibpdVNkFDYC&pg=PA3&dq=%22an+algebra+over+a+field+k%22">3</a> Proposition 1.1.1</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><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 id="CITEREFProlla2011" class="citation book cs1">Prolla, João B. (2011) [1977]. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=utTS4nTd-IsC">"Lemma 4.10"</a>. <i>Approximation of Vector Valued Functions</i>. Elsevier. p. 65. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-08-087136-3" title="Special:BookSources/978-0-08-087136-3"><bdi>978-0-08-087136-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Lemma+4.10&rft.btitle=Approximation+of+Vector+Valued+Functions&rft.pages=65&rft.pub=Elsevier&rft.date=2011&rft.isbn=978-0-08-087136-3&rft.aulast=Prolla&rft.aufirst=Jo%C3%A3o+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DutTS4nTd-IsC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebra+over+a+field" class="Z3988"></span></span> </li> <li id="cite_note-Schafer-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Schafer_3-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchafer1996" class="citation book cs1"><a href="/wiki/Richard_D._Schafer" title="Richard D. Schafer">Schafer, Richard D.</a> (1996). <a rel="nofollow" class="external text" href="http://www.gutenberg.org/ebooks/25156"><i>An Introduction to Nonassociative Algebras</i></a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-68813-5" title="Special:BookSources/0-486-68813-5"><bdi>0-486-68813-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Nonassociative+Algebras&rft.date=1996&rft.isbn=0-486-68813-5&rft.aulast=Schafer&rft.aufirst=Richard+D.&rft_id=http%3A%2F%2Fwww.gutenberg.org%2Febooks%2F25156&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebra+over+a+field" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStudy1890" class="citation cs2">Study, E. (1890), "Über Systeme complexer Zahlen und ihre Anwendungen in der Theorie der Transformationsgruppen", <i>Monatshefte für Mathematik</i>, <b>1</b> (1): 283–354, <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%2FBF01692479">10.1007/BF01692479</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:121426669">121426669</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Monatshefte+f%C3%BCr+Mathematik&rft.atitle=%C3%9Cber+Systeme+complexer+Zahlen+und+ihre+Anwendungen+in+der+Theorie+der+Transformationsgruppen&rft.volume=1&rft.issue=1&rft.pages=283-354&rft.date=1890&rft_id=info%3Adoi%2F10.1007%2FBF01692479&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121426669%23id-name%3DS2CID&rft.aulast=Study&rft.aufirst=E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebra+over+a+field" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatsumura1989" class="citation book cs1">Matsumura, H. (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yJwNrABugDEC"><i>Commutative Ring Theory</i></a>. Cambridge Studies in Advanced Mathematics. Vol. 8. Translated by Reid, M. (2nd ed.). Cambridge University Press. <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=H.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DyJwNrABugDEC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebra+over+a+field" class="Z3988"></span><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (November 2024)">page needed</span></a></i>]</sup></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKunz1985" class="citation book cs1">Kunz, Ernst (1985). <i>Introduction to Commutative algebra and algebraic geometry</i>. Birkhauser. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8176-3065-1" title="Special:BookSources/0-8176-3065-1"><bdi>0-8176-3065-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Commutative+algebra+and+algebraic+geometry&rft.pub=Birkhauser&rft.date=1985&rft.isbn=0-8176-3065-1&rft.aulast=Kunz&rft.aufirst=Ernst&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebra+over+a+field" class="Z3988"></span><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources"><span title="This citation requires a reference to the specific page or range of pages in which the material appears. (November 2024)">page needed</span></a></i>]</sup></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Algebra_over_a_field&action=edit&section=20" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHazewinkelGubareniKirichenko2004" class="citation book cs1"><a href="/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Hazewinkel, Michiel</a>; Gubareni, Nadiya; Kirichenko, Vladimir V. (2004). <i>Algebras, rings and modules</i>. Vol. 1. Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4020-2690-0" title="Special:BookSources/1-4020-2690-0"><bdi>1-4020-2690-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebras%2C+rings+and+modules&rft.pub=Springer&rft.date=2004&rft.isbn=1-4020-2690-0&rft.aulast=Hazewinkel&rft.aufirst=Michiel&rft.au=Gubareni%2C+Nadiya&rft.au=Kirichenko%2C+Vladimir+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AAlgebra+over+a+field" class="Z3988"></span></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Algebra_over_a_field&oldid=1258206648">https://en.wikipedia.org/w/index.php?title=Algebra_over_a_field&oldid=1258206648</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Algebras" title="Category:Algebras">Algebras</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Wikipedia_articles_needing_page_number_citations_from_November_2024" title="Category:Wikipedia articles needing page number citations from November 2024">Wikipedia articles needing page number citations from November 2024</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 18 November 2024, at 18:45<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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