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vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Extension_and_contraction_of_an_ideal"> <div class="vector-toc-text"> 8 Extension and contraction of an ideal </div> </a> <ul id="toc-Extension_and_contraction_of_an_ideal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalizations" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalizations"> <div class="vector-toc-text"> 9 Generalizations </div> </a> <ul id="toc-Generalizations-sublist" class="vector-toc-list"> </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"> 10 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 11 Notes </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"> 12 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 13 External links </div> </a> <ul id="toc-External_links-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 aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Ideal (ring theory)</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" 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Available in 40 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-40" aria-hidden="true" > 40 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%AB%D8%A7%D9%84%D9%8A_(%D9%86%D8%B8%D8%B1%D9%8A%D8%A9_%D8%A7%D9%84%D8%AD%D9%84%D9%82%D8%A7%D8%AA)" title="مثالي (نظرية الحلقات) – Arabic" lang="ar" hreflang="ar" data-title="مثالي (نظرية الحلقات)" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%98%D0%B4%D0%B5%D0%B0%D0%BB_(%D1%82%D0%B5%D0%BE%D1%80%D0%B8%D1%8F_%D0%BD%D0%B0_%D0%BF%D1%80%D1%8A%D1%81%D1%82%D0%B5%D0%BD%D0%B8%D1%82%D0%B5)" title="Идеал (теория на пръстените) – Bulgarian" lang="bg" hreflang="bg" data-title="Идеал (теория на пръстените)" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Ideal_(matem%C3%A0tiques)" title="Ideal (matemàtiques) – Catalan" lang="ca" hreflang="ca" data-title="Ideal (matemàtiques)" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%98%D0%B4%D0%B5%D0%B0%D0%BB_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Идеал (алгебра) – Chuvash" lang="cv" hreflang="cv" data-title="Идеал (алгебра)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Ide%C3%A1l_(teorie_okruh%C5%AF)" title="Ideál (teorie okruhů) – Czech" lang="cs" hreflang="cs" data-title="Ideál (teorie okruhů)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Ideal_(ringteori)" title="Ideal (ringteori) – Danish" lang="da" hreflang="da" data-title="Ideal (ringteori)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ideal_(Ringtheorie)" title="Ideal (Ringtheorie) – German" lang="de" hreflang="de" data-title="Ideal (Ringtheorie)" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ringi_ideaal" title="Ringi ideaal – Estonian" lang="et" hreflang="et" data-title="Ringi ideaal" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%99%CE%B4%CE%B5%CF%8E%CE%B4%CE%B5%CF%82_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Ιδεώδες (μαθηματικά) – Greek" lang="el" hreflang="el" data-title="Ιδεώδες (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ideal_(teor%C3%ADa_de_anillos)" title="Ideal (teoría de anillos) – Spanish" lang="es" hreflang="es" data-title="Ideal (teoría de anillos)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Idealo_(algebro)" title="Idealo (algebro) – Esperanto" lang="eo" hreflang="eo" data-title="Idealo (algebro)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%A7%DB%8C%D8%AF%D9%87%E2%80%8C%D8%A2%D9%84_(%D9%86%D8%B8%D8%B1%DB%8C%D9%87_%D8%AD%D9%84%D9%82%D9%87%E2%80%8C%D9%87%D8%A7)" title="ایده‌آل (نظریه حلقه‌ها) – Persian" lang="fa" hreflang="fa" data-title="ایده‌آل (نظریه حلقه‌ها)" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Id%C3%A9al" title="Idéal – French" lang="fr" hreflang="fr" data-title="Idéal" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Ideal_(teor%C3%ADa_dos_aneis)" title="Ideal (teoría dos aneis) – Galician" lang="gl" hreflang="gl" data-title="Ideal (teoría dos aneis)" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%95%84%EC%9D%B4%EB%94%94%EC%96%BC" title="아이디얼 – Korean" lang="ko" hreflang="ko" data-title="아이디얼" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Ideal_(teori_gelanggang)" title="Ideal (teori gelanggang) – Indonesian" lang="id" hreflang="id" data-title="Ideal (teori gelanggang)" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Ideal_(theoria_de_anellos)" title="Ideal (theoria de anellos) – Interlingua" lang="ia" hreflang="ia" data-title="Ideal (theoria de anellos)" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Ideale_(matematica)" title="Ideale (matematica) – Italian" lang="it" hreflang="it" data-title="Ideale (matematica)" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%99%D7%93%D7%99%D7%90%D7%9C_(%D7%90%D7%9C%D7%92%D7%91%D7%A8%D7%94)" title="אידיאל (אלגברה) – Hebrew" lang="he" hreflang="he" data-title="אידיאל (אלגברה)" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%98%D0%B4%D0%B5%D0%B0%D0%BB_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%B4%D0%B0)" title="Идеал (математикада) – Kazakh" lang="kk" hreflang="kk" data-title="Идеал (математикада)" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Ideaal_(matem%C3%A0tica)" title="Ideaal (matemàtica) – Lombard" lang="lmo" hreflang="lmo" data-title="Ideaal (matemàtica)" data-language-autonym="Lombard" data-language-local-name="Lombard" class="interlanguage-link-target">Lombard</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Ide%C3%A1l_(gy%C5%B1r%C5%B1elm%C3%A9let)" title="Ideál (gyűrűelmélet) – Hungarian" lang="hu" hreflang="hu" data-title="Ideál (gyűrűelmélet)" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Ideaal_(ringtheorie)" title="Ideaal (ringtheorie) – Dutch" lang="nl" hreflang="nl" data-title="Ideaal (ringtheorie)" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%A4%E3%83%87%E3%82%A2%E3%83%AB_(%E7%92%B0%E8%AB%96)" title="イデアル (環論) – Japanese" lang="ja" hreflang="ja" data-title="イデアル (環論)" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Ideal_(matematikk)" title="Ideal (matematikk) – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Ideal (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Ideal_i_matematikk" title="Ideal i matematikk – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Ideal i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Idea%C5%82_(teoria_pier%C5%9Bcieni)" title="Ideał (teoria pierścieni) – Polish" lang="pl" hreflang="pl" data-title="Ideał (teoria pierścieni)" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Ideal_(teoria_dos_an%C3%A9is)" title="Ideal (teoria dos anéis) – Portuguese" lang="pt" hreflang="pt" data-title="Ideal (teoria dos anéis)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Ideal_(teoria_inelelor)" title="Ideal (teoria inelelor) – Romanian" lang="ro" hreflang="ro" data-title="Ideal (teoria inelelor)" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%98%D0%B4%D0%B5%D0%B0%D0%BB_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Идеал (алгебра) – Russian" lang="ru" hreflang="ru" data-title="Идеал (алгебра)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Ide%C3%A1l_(okruhu)" title="Ideál (okruhu) – Slovak" lang="sk" hreflang="sk" data-title="Ideál (okruhu)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Ideal_(teorija_kolobarjev)" title="Ideal (teorija kolobarjev) – Slovenian" lang="sl" hreflang="sl" data-title="Ideal (teorija kolobarjev)" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%B4%D0%B5%D0%B0%D0%BB_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Идеал (математика) – Serbian" lang="sr" hreflang="sr" data-title="Идеал (математика)" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ideaali_(rengasteoria)" title="Ideaali (rengasteoria) – Finnish" lang="fi" hreflang="fi" data-title="Ideaali (rengasteoria)" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ideal_(ringteori)" title="Ideal (ringteori) – Swedish" lang="sv" hreflang="sv" data-title="Ideal (ringteori)" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%B5%E0%AE%B3%E0%AF%88%E0%AE%AF%E0%AE%A4%E0%AF%8D%E0%AE%A4%E0%AE%BF%E0%AE%B2%E0%AF%8D_%E0%AE%9A%E0%AF%80%E0%AE%B0%E0%AF%8D%E0%AE%AE%E0%AE%AE%E0%AF%8D_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="வளையத்தில் சீர்மம் (கணிதம்) – Tamil" lang="ta" hreflang="ta" data-title="வளையத்தில் சீர்மம் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%86%D0%B4%D0%B5%D0%B0%D0%BB_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Ідеал (алгебра) – Ukrainian" lang="uk" hreflang="uk" data-title="Ідеал (алгебра)" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/I-%C4%91%C3%AA-an" title="I-đê-an – Vietnamese" lang="vi" hreflang="vi" data-title="I-đê-an" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%92%B0%E5%80%8D%E6%95%B8" title="環倍數 – Cantonese" lang="yue" hreflang="yue" data-title="環倍數" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target">粵語</a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%90%86%E6%83%B3_(%E7%8E%AF%E8%AE%BA)" title="理想 (环论) – Chinese" lang="zh" hreflang="zh" data-title="理想 (环论)" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target">中文</a></li> </ul> <div class="after-portlet after-portlet-lang"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q44649#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Ideal_(ring_theory)" title="View the content page [c]" accesskey="c">Article</a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Ideal_(ring_theory)" rel="discussion" title="Discuss improvements to the content page [t]" 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.sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist" style="width: 20.5em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → Ring theory <a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a></th></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)">Basic concepts</div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Rings</a> <dl><dd>• <a href="/wiki/Subring" title="Subring">Subrings</a></dd> <dd>• <a class="mw-selflink selflink">Ideal</a></dd> <dd>• <a href="/wiki/Quotient_ring" title="Quotient ring">Quotient ring</a> <dl><dd>• <a href="/wiki/Fractional_ideal" title="Fractional ideal">Fractional ideal</a></dd> <dd>• <a href="/wiki/Total_ring_of_fractions" title="Total ring of fractions">Total ring of fractions</a></dd></dl></dd> <dd>• <a href="/wiki/Product_of_rings" title="Product of rings">Product of rings</a></dd> <dd>• <a href="/wiki/Free_product_of_associative_algebras" title="Free product of associative algebras">Free product of associative algebras</a></dd> <dd>• <a href="/wiki/Tensor_product_of_algebras" title="Tensor product of algebras">Tensor product of algebras</a></dd></dl> <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">Ring homomorphisms</a> <dl><dd>• <a href="/wiki/Kernel_(algebra)#Ring_homomorphisms" title="Kernel (algebra)">Kernel</a></dd> <dd>• <a href="/wiki/Inner_automorphism#Ring_case" title="Inner automorphism">Inner automorphism</a></dd> <dd>• <a href="/wiki/Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius endomorphism</a></dd></dl> <a href="/wiki/Algebraic_structure" title="Algebraic structure">Algebraic structures</a> <dl><dd>• <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">Module</a></dd> <dd>• <a href="/wiki/Associative_algebra" title="Associative algebra">Associative algebra</a></dd> <dd>• <a href="/wiki/Graded_ring" title="Graded ring">Graded ring</a></dd> <dd>• <a href="/wiki/Involutive_ring" class="mw-redirect" title="Involutive ring">Involutive ring</a></dd> <dd>• <a href="/wiki/Category_of_rings" title="Category of rings">Category of rings</a> <dl><dd>• <a href="/wiki/Integer" title="Integer">Initial ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></dd> <dd>• <a href="/wiki/Zero_ring" title="Zero ring">Terminal ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45ab495cb8cfbac68a9322af662c3d6c7dbe494" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.686ex; height:2.843ex;" alt="{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }"></dd></dl></dd></dl> Related structures <dl><dd>• <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> <dl><dd>• <a href="/wiki/Finite_field" title="Finite field">Finite field</a></dd></dl></dd> <dd>• <a href="/wiki/Non-associative_ring" class="mw-redirect" title="Non-associative ring">Non-associative ring</a> <dl><dd>• <a href="/wiki/Lie_ring" class="mw-redirect" title="Lie ring">Lie ring</a></dd> <dd>• <a href="/wiki/Jordan_ring" class="mw-redirect" title="Jordan ring">Jordan ring</a></dd></dl></dd> <dd>• <a href="/wiki/Semiring" title="Semiring">Semiring</a> <dl><dd>• <a href="/wiki/Semifield" title="Semifield">Semifield</a></dd></dl></dd></dl></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="/wiki/Commutative_ring" title="Commutative ring">Commutative rings</a> <dl><dd>• <a href="/wiki/Integral_domain" title="Integral domain">Integral domain</a> <dl><dd>• <a href="/wiki/Integrally_closed_domain" title="Integrally closed domain">Integrally closed domain</a></dd> <dd>• <a href="/wiki/GCD_domain" title="GCD domain">GCD domain</a></dd> <dd>• <a href="/wiki/Unique_factorization_domain" title="Unique factorization domain">Unique factorization domain</a></dd> <dd>• <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">Principal ideal domain</a></dd> <dd>• <a href="/wiki/Euclidean_domain" title="Euclidean domain">Euclidean domain</a></dd> <dd>• <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a> <dl><dd>• <a href="/wiki/Finite_field" title="Finite field">Finite field</a></dd></dl></dd> <dd>• <a href="/wiki/Polynomial_ring" title="Polynomial ring">Polynomial ring</a></dd> <dd>• <a href="/wiki/Formal_power_series_ring" class="mw-redirect" title="Formal power series ring">Formal power series ring</a></dd></dl></dd></dl> <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a> <dl><dd>• <a href="/wiki/Algebraic_number_field" title="Algebraic number field">Algebraic number field</a></dd> <dd>• <a href="/wiki/Integers_modulo_n" class="mw-redirect" title="Integers modulo n">Integers modulo n</a></dd> <dd>• <a href="/wiki/Ring_of_integers" title="Ring of integers">Ring of integers</a></dd> <dd>• <a href="/wiki/P-adic_integer" class="mw-redirect" title="P-adic integer">p-adic integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}"></dd> <dd>• <a href="/wiki/P-adic_number" title="P-adic number">p-adic numbers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Q} _{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f44bc6894c682710705f3ea74f33042e0acc3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}"></dd> <dd>• <a href="/wiki/Pr%C3%BCfer_group#The_Prüfer_group_as_a_ring" title="Prüfer group">Prüfer p-ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} (p^{\infty })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} (p^{\infty })}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14af623e08c241266c125ad927dd35086ec8ce90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.404ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} (p^{\infty })}"></dd></dl></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Noncommutative_algebra" class="mw-redirect" title="Noncommutative algebra">Noncommutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><a href="/wiki/Noncommutative_ring" title="Noncommutative ring">Noncommutative rings</a> <dl><dd>• <a href="/wiki/Division_ring" title="Division ring">Division ring</a></dd> <dd>• <a href="/wiki/Semiprimitive_ring" title="Semiprimitive ring">Semiprimitive ring</a></dd> <dd>• <a href="/wiki/Simple_ring" title="Simple ring">Simple ring</a></dd> <dd>• <a href="/wiki/Commutator_(ring_theory)" class="mw-redirect" title="Commutator (ring theory)">Commutator</a></dd></dl> <a href="/wiki/Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a> <a href="/wiki/Free_algebra" title="Free algebra">Free algebra</a> <a href="/wiki/Clifford_algebra" title="Clifford algebra">Clifford algebra</a> <dl><dd>• <a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></dd></dl> <a href="/wiki/Operator_algebra" title="Operator algebra">Operator algebra</a></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Ring_theory_sidebar" title="Template:Ring theory sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Ring_theory_sidebar" title="Template talk:Ring theory sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Ring_theory_sidebar" title="Special:EditPage/Template:Ring theory sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, and more specifically in <a href="/wiki/Ring_theory" title="Ring theory">ring theory</a>, an ideal of a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> is a special <a href="/wiki/Subset" title="Subset">subset</a> of its elements. Ideals generalize certain subsets of the <a href="/wiki/Integer" title="Integer">integers</a>, such as the <a href="/wiki/Even_numbers" class="mw-redirect" title="Even numbers">even numbers</a> or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closure</a> and absorption properties are the defining properties of an ideal. An ideal can be used to construct a <a href="/wiki/Quotient_ring" title="Quotient ring">quotient ring</a> in a way similar to how, in <a href="/wiki/Group_theory" title="Group theory">group theory</a>, a <a href="/wiki/Normal_subgroup" title="Normal subgroup">normal subgroup</a> can be used to construct a <a href="/wiki/Quotient_group" title="Quotient group">quotient group</a>. Among the integers, the ideals correspond one-for-one with the <a href="/wiki/Non-negative_integer" class="mw-redirect" title="Non-negative integer">non-negative integers</a>: in this ring, every ideal is a <a href="/wiki/Principal_ideal" title="Principal ideal">principal ideal</a> consisting of the multiples of a single non-negative number. However, in other rings, the ideals may not correspond directly to the ring elements, and certain properties of integers, when generalized to rings, attach more naturally to the ideals than to the elements of the ring. For instance, the <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideals</a> of a ring are analogous to <a href="/wiki/Prime_number" title="Prime number">prime numbers</a>, and the <a href="/wiki/Chinese_remainder_theorem" title="Chinese remainder theorem">Chinese remainder theorem</a> can be generalized to ideals. There is a version of <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">unique prime factorization</a> for the ideals of a <a href="/wiki/Dedekind_domain" title="Dedekind domain">Dedekind domain</a> (a type of ring important in <a href="/wiki/Number_theory" title="Number theory">number theory</a>). The related, but distinct, concept of an <a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">ideal</a> in <a href="/wiki/Order_theory" title="Order theory">order theory</a> is derived from the notion of ideal in ring theory. A <a href="/wiki/Fractional_ideal" title="Fractional ideal">fractional ideal</a> is a generalization of an ideal, and the usual ideals are sometimes called integral ideals for clarity. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Ideal_(ring_theory)&action=edit&section=1" title="Edit section: History">edit</a>]</div> <a href="/wiki/Ernst_Kummer" title="Ernst Kummer">Ernst Kummer</a> invented the concept of <a href="/wiki/Ideal_number" title="Ideal number">ideal numbers</a> to serve as the "missing" factors in number rings in which unique factorization fails; here the word "ideal" is in the sense of existing in imagination only, in analogy with "ideal" objects in geometry such as points at infinity.<a href="#cite_note-Stillwell-1">[1]</a> In 1876, <a href="/wiki/Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> replaced Kummer's undefined concept by concrete sets of numbers, sets that he called ideals, in the third edition of <a href="/wiki/Dirichlet" class="mw-redirect" title="Dirichlet">Dirichlet</a>'s book <a href="/wiki/Vorlesungen_%C3%BCber_Zahlentheorie" title="Vorlesungen über Zahlentheorie">Vorlesungen über Zahlentheorie</a>, to which Dedekind had added many supplements.<a href="#cite_note-Stillwell-1">[1]</a><a href="#cite_note-flt-2">[2]</a><a href="#cite_note-everest_ward-3">[3]</a> Later the notion was extended beyond number rings to the setting of polynomial rings and other commutative rings by <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> and especially <a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a>. <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2>[<a href="/w/index.php?title=Ideal_(ring_theory)&action=edit&section=2" title="Edit section: Definitions">edit</a>]</div> Given a <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> R, a left ideal is a subset I of R that is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the <a href="/wiki/Additive_group" title="Additive group">additive group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> that "absorbs multiplication from the left by elements of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">⁠"; that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is a left ideal if it satisfies the following two conditions: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (I,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>I</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (I,+)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9795f78cf420c87df00914f8dfad7047db1fa807" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.823ex; height:2.843ex;" alt="{\displaystyle (I,+)}"> is a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (R,+)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mo>+</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (R,+)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/466356a631ac93bc70fbe2d276117d22f980e285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.415ex; height:2.843ex;" alt="{\displaystyle (R,+)}">⁠,</li> <li>For every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca49c66b5e9b5f32249a737e4429c3df136c33f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.653ex; height:2.176ex;" alt="{\displaystyle r\in R}"> and every ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dec8caa8f241cb38a5348d7937b538227ad32c48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.342ex; height:2.176ex;" alt="{\displaystyle x\in I}">⁠, the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10279e90b103d9755ac5fb24ba474d8ce6ec133f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.378ex; height:1.676ex;" alt="{\displaystyle rx}"> is in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}">⁠.<a href="#cite_note-4">[4]</a></li></ol> In other words, a left ideal is a left <a href="/wiki/Submodule" class="mw-redirect" title="Submodule">submodule</a> of R, considered as a <a href="/wiki/Left_module" class="mw-redirect" title="Left module">left module</a> over itself.<a href="#cite_note-5">[5]</a> A right ideal is defined similarly, with the condition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rx\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>x</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rx\in I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eda8509b3e70011a622c1665430563036cf16cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.391ex; height:2.176ex;" alt="{\displaystyle rx\in I}"> replaced by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xr\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>r</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xr\in I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2142379ff5c6f63d3be75f988d477050a59fde87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.391ex; height:2.176ex;" alt="{\displaystyle xr\in I}">⁠. A two-sided ideal is a left ideal that is also a right ideal. If the ring is <a href="/wiki/Commutative_ring" title="Commutative ring">commutative</a>, the three definitions are the same, and one talks simply of an ideal. In the non-commutative case, "ideal" is often used instead of "two-sided ideal". If I is a left, right or two-sided ideal, the relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∼</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\sim y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd1014d850b7c883eb76301dd58c643e3c7e4eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x\sim y}"> if and only if <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-y\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-y\in I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a38120b4e5728ed34204ea954872e3dcf828749" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.338ex; height:2.509ex;" alt="{\displaystyle x-y\in I}"></dd></dl> is an <a href="/wiki/Equivalence_relation" title="Equivalence relation">equivalence relation</a> on R, and the set of <a href="/wiki/Equivalence_class" title="Equivalence class">equivalence classes</a> forms a left, right or bi module denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R/I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R/I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0650c975ee7bf3b39ce3144a5de71179c40ee493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.098ex; height:2.843ex;" alt="{\displaystyle R/I}"> and called the <a href="/wiki/Quotient_module" title="Quotient module">quotient</a> of R by I.<a href="#cite_note-6">[6]</a> (It is an instance of a <a href="/wiki/Congruence_relation" title="Congruence relation">congruence relation</a> and is a generalization of <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a>.) If the ideal I is two-sided, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R/I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R/I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0650c975ee7bf3b39ce3144a5de71179c40ee493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.098ex; height:2.843ex;" alt="{\displaystyle R/I}"> is a ring,<a href="#cite_note-7">[7]</a> and the function <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\to R/I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">→</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\to R/I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec2da9cf0094c5948d4b0b56d8132683d902f7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.476ex; height:2.843ex;" alt="{\displaystyle R\to R/I}"></dd></dl> that associates to each element of R its equivalence class is a <a href="/wiki/Surjective" class="mw-redirect" title="Surjective">surjective</a> <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> that has the ideal as its <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)">kernel</a>.<a href="#cite_note-8">[8]</a> Conversely, the kernel of a ring homomorphism is a two-sided ideal. Therefore, the two-sided ideals are exactly the kernels of ring homomorphisms. <div class="mw-heading mw-heading3"><h3 id="Note_on_convention">Note on convention</h3>[<a href="/w/index.php?title=Ideal_(ring_theory)&action=edit&section=3" title="Edit section: Note on convention">edit</a>]</div> By convention, a ring has the multiplicative identity. But some authors do not require a ring to have the multiplicative identity; i.e., for them, a ring is a <a href="/wiki/Rng_(mathematics)" class="mw-redirect" title="Rng (mathematics)">rng</a>. For a rng R, a left ideal I is a subrng with the additional property that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle rx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10279e90b103d9755ac5fb24ba474d8ce6ec133f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.378ex; height:1.676ex;" alt="{\displaystyle rx}"> is in I for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca49c66b5e9b5f32249a737e4429c3df136c33f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.653ex; height:2.176ex;" alt="{\displaystyle r\in R}"> and every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dec8caa8f241cb38a5348d7937b538227ad32c48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.342ex; height:2.176ex;" alt="{\displaystyle x\in I}">. (Right and two-sided ideals are defined similarly.) For a ring, an ideal I (say a left ideal) is rarely a subring; since a subring shares the same multiplicative identity with the ambient ring R, if I were a subring, for every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca49c66b5e9b5f32249a737e4429c3df136c33f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.653ex; height:2.176ex;" alt="{\displaystyle r\in R}">, we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=r1\in I;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mi>r</mi> <mn>1</mn> <mo>∈</mo> <mi>I</mi> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=r1\in I;}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad71212c9ec41171bd6ee2b54cc4d866e53da8d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.017ex; height:2.509ex;" alt="{\displaystyle r=r1\in I;}"> i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ed3470fadf888a1e4f3a018c699e78de45501c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.034ex; height:2.176ex;" alt="{\displaystyle I=R}">. The notion of an ideal does not involve associativity; thus, an ideal is also defined for <a href="/wiki/Non-associative_ring" class="mw-redirect" title="Non-associative ring">non-associative rings</a> (often without the multiplicative identity) such as a <a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a>. <div class="mw-heading mw-heading2"><h2 id="Examples_and_properties">Examples and properties</h2>[<a href="/w/index.php?title=Ideal_(ring_theory)&action=edit&section=4" title="Edit section: Examples and properties">edit</a>]</div> (For the sake of brevity, some results are stated only for left ideals but are usually also true for right ideals with appropriate notation changes.) <ul><li>In a ring R, the set R itself forms a two-sided ideal of R called the unit ideal. It is often also denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a25115739469707c4758b189fe310a750092a80a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (1)}"> since it is precisely the two-sided ideal generated (see below) by the unity ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{R}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{R}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4b84310fba9b328ce04e95ee5aeaf1c81236785" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.642ex; height:2.509ex;" alt="{\displaystyle 1_{R}}">⁠. Also, the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0_{R}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0_{R}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07a13e6e1ad75eed0eb036d9b14134133fd86291" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.967ex; height:2.843ex;" alt="{\displaystyle \{0_{R}\}}"> consisting of only the additive identity 0R forms a two-sided ideal called the zero ideal and is denoted by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf6d00d0d700cc9a652fb7e4b7f858b6eb2e9925" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle (0)}">⁠.<a href="#cite_note-9">[note 1]</a> Every (left, right or two-sided) ideal contains the zero ideal and is contained in the unit ideal.<a href="#cite_note-FOOTNOTEDummitFoote2004243-10">[9]</a></li> <li>An (left, right or two-sided) ideal that is not the unit ideal is called a proper ideal (as it is a <a href="/wiki/Proper_subset" class="mw-redirect" title="Proper subset">proper subset</a>).<a href="#cite_note-11">[10]</a> Note: a left ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}"> is proper if and only if it does not contain a unit element, since if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\in {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\in {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5665b13bc4f45deb850d3f1d346aa200f550901" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.333ex; height:1.843ex;" alt="{\displaystyle u\in {\mathfrak {a}}}"> is a unit element, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=(ru^{-1})u\in {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>r</mi> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mi>u</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r=(ru^{-1})u\in {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ca96b60bb0f20cee6fcf83d25b02df5d873401b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16ex; height:3.176ex;" alt="{\displaystyle r=(ru^{-1})u\in {\mathfrak {a}}}"> for every ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca49c66b5e9b5f32249a737e4429c3df136c33f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.653ex; height:2.176ex;" alt="{\displaystyle r\in R}">⁠. Typically there are plenty of proper ideals. In fact, if R is a <a href="/wiki/Skew-field" class="mw-redirect" title="Skew-field">skew-field</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0),(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0),(1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cfc8d3a91c71a3be47618191b39bc6c6584d328" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.977ex; height:2.843ex;" alt="{\displaystyle (0),(1)}"> are its only ideals and conversely: that is, a nonzero ring R is a skew-field if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0),(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0),(1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cfc8d3a91c71a3be47618191b39bc6c6584d328" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.977ex; height:2.843ex;" alt="{\displaystyle (0),(1)}"> are the only left (or right) ideals. (Proof: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> is a nonzero element, then the principal left ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Rx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Rx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8072625515d9f43b2fd354deb2126199df9bb4cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.094ex; height:2.176ex;" alt="{\displaystyle Rx}"> (see below) is nonzero and thus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Rx=(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>x</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Rx=(1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acb14b60cc8a52a2259b857ff42301f46ee464c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.164ex; height:2.843ex;" alt="{\displaystyle Rx=(1)}">; i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle yx=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mi>x</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle yx=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f345b54e1fe329ebb23ee5ec275aef5fdb67724" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.746ex; height:2.509ex;" alt="{\displaystyle yx=1}"> for some nonzero ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}">⁠. Likewise, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle zy=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle zy=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43bc5f5a726623f0a722be8f0f947d2641e3834e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.505ex; height:2.509ex;" alt="{\displaystyle zy=1}"> for some nonzero <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=z(yx)=(zy)x=x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>z</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo>=</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=z(yx)=(zy)x=x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b3dbfec9677c9123272d31fd933e12f5c548013" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.478ex; height:2.843ex;" alt="{\displaystyle z=z(yx)=(zy)x=x}">.)</li> <li>The even <a href="/wiki/Integer" title="Integer">integers</a> form an ideal in the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"> of all integers, since the sum of any two even integers is even, and the product of any integer with an even integer is also even; this ideal is usually denoted by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c112372c16743e8c02db5eac5c6de0659841bec6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle 2\mathbb {Z} }">⁠. More generally, the set of all integers divisible by a fixed integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is an ideal denoted ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33e5e8bd9b24ba5d0b31eda653b78a63e1af831d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.945ex; height:2.176ex;" alt="{\displaystyle n\mathbb {Z} }">⁠. In fact, every non-zero ideal of the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"> is generated by its smallest positive element, as a consequence of <a href="/wiki/Euclidean_division" title="Euclidean division">Euclidean division</a>, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"> is a <a href="/wiki/Principal_ideal_domain" title="Principal ideal domain">principal ideal domain</a>.<a href="#cite_note-FOOTNOTEDummitFoote2004243-10">[9]</a></li> <li>The set of all <a href="/wiki/Polynomial" title="Polynomial">polynomials</a> with real coefficients that are divisible by the polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{2}+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92a3a8d23f9f8123651e496dcf8490990c65cf9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.387ex; height:2.843ex;" alt="{\displaystyle x^{2}+1}"> is an ideal in the ring of all real-coefficient polynomials ⁠<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><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/453d1013f9dd290be70d5fe534e0d3311b0a7c6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.301ex; height:2.843ex;" alt="{\displaystyle \mathbb {R} [x]}">⁠.</li> <li>Take a ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> and positive integer ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">⁠. For each ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq i\leq n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abbe58b9b83f8b6ec0da570e2249323a8930ef1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.557ex; height:2.343ex;" alt="{\displaystyle 1\leq i\leq n}">⁠, the set of all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a> with entries in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> whose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">-th row is zero is a right ideal in the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{n}(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M_{n}(R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/096af14d3445ce0171d03e59a6ac5d8a70317489" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.046ex; height:2.843ex;" alt="{\displaystyle M_{n}(R)}"> of all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> matrices with entries in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">⁠. It is not a left ideal. Similarly, for each ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq j\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>j</mi> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq j\leq n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/829066d1e818107baff629179da368c45fcb8883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.712ex; height:2.509ex;" alt="{\displaystyle 1\leq j\leq n}">⁠, the set of all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> matrices whose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}">-th column is zero is a left ideal but not a right ideal.</li> <li>The ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c83d1af403f8fe4ad3b88bf35451369497d8c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.254ex; height:2.843ex;" alt="{\displaystyle C(\mathbb {R} )}"> of all <a href="/wiki/Continuous_function" title="Continuous function">continuous functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }"> under <a href="/wiki/Pointwise_multiplication" class="mw-redirect" title="Pointwise multiplication">pointwise multiplication</a> contains the ideal of all continuous functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> such that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(1)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(1)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da761295ab324acdb04e33ab75299c7fb588675f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\displaystyle f(1)=0}">⁠.<a href="#cite_note-FOOTNOTEDummitFoote2004244-12">[11]</a> Another ideal in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C(\mathbb {R} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C(\mathbb {R} )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7c83d1af403f8fe4ad3b88bf35451369497d8c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.254ex; height:2.843ex;" alt="{\displaystyle C(\mathbb {R} )}"> is given by those functions that vanish for large enough arguments, i.e. those continuous functions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> for which there exists a number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L>0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L>0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78d06bfbe00ff463870f868c958b37cbe46ea3a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.844ex; height:2.176ex;" alt="{\displaystyle L>0}"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf85883d74b75fe35ca8d3f2b44802df078e4fa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.678ex; height:2.843ex;" alt="{\displaystyle f(x)=0}"> whenever ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vert x\vert >L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">|</mo> <mi>x</mi> <mo fence="false" stretchy="false">|</mo> <mo>></mo> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vert x\vert >L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f14f778aaf586ed4b9e6f6e297042233a5dc9077" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.305ex; height:2.843ex;" alt="{\displaystyle \vert x\vert >L}">⁠.</li> <li>A ring is called a <a href="/wiki/Simple_ring" title="Simple ring">simple ring</a> if it is nonzero and has no two-sided ideals other than ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0),(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0),(1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cfc8d3a91c71a3be47618191b39bc6c6584d328" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.977ex; height:2.843ex;" alt="{\displaystyle (0),(1)}">⁠. Thus, a skew-field is simple and a simple commutative ring is a field. The <a href="/wiki/Matrix_ring" title="Matrix ring">matrix ring</a> over a skew-field is a simple ring.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:R\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>R</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:R\to S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9aa7daf2186fbf2fb3b5090e495ad432d4cf42c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.093ex; height:2.509ex;" alt="{\displaystyle f:R\to S}"> is a <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a>, then the kernel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker(f)=f^{-1}(0_{S})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ker(f)=f^{-1}(0_{S})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6878229c8d5cfcc7ab1191358bb741c39697e854" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.275ex; height:3.176ex;" alt="{\displaystyle \ker(f)=f^{-1}(0_{S})}"> is a two-sided ideal of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">⁠.<a href="#cite_note-FOOTNOTEDummitFoote2004243-10">[9]</a> By definition, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(1_{R})=1_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(1_{R})=1_{S}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c922d3c4a7aa35a9703521f921971b0e8fa7832" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.283ex; height:2.843ex;" alt="{\displaystyle f(1_{R})=1_{S}}">⁠, and thus if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is not the zero ring (so ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1_{S}\neq 0_{S}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo>≠</mo> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{S}\neq 0_{S}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64d7b267c7e76856f3890efa674cfd6264688110" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.008ex; height:2.676ex;" alt="{\displaystyle 1_{S}\neq 0_{S}}">⁠), then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ker(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ker</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ker(f)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39d2eca3185b45d4462e15bf85bd4ad36e32062d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.259ex; height:2.843ex;" alt="{\displaystyle \ker(f)}"> is a proper ideal. More generally, for each left ideal I of S, the pre-image <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}(I)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a8452df1221e90eed16265d61c0295ba54f457d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.634ex; height:3.176ex;" alt="{\displaystyle f^{-1}(I)}"> is a left ideal. If I is a left ideal of R, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(I)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1120c98155741515143b09d7265a1d394fcf003d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.26ex; height:2.843ex;" alt="{\displaystyle f(I)}"> is a left ideal of the subring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/482b1cec7d8f410485628cf860c3b063c3cc9f59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.852ex; height:2.843ex;" alt="{\displaystyle f(R)}"> of S: unless f is surjective, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(I)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1120c98155741515143b09d7265a1d394fcf003d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.26ex; height:2.843ex;" alt="{\displaystyle f(I)}"> need not be an ideal of S; see also <a href="#Extension_and_contraction_of_an_ideal">#Extension and contraction of an ideal</a> below.</li> <li>Ideal correspondence: Given a surjective ring homomorphism ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:R\to S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>:</mo> <mi>R</mi> <mo stretchy="false">→</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f:R\to S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9aa7daf2186fbf2fb3b5090e495ad432d4cf42c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.093ex; height:2.509ex;" alt="{\displaystyle f:R\to S}">⁠, there is a bijective order-preserving correspondence between the left (resp. right, two-sided) ideals of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> containing the kernel of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}"> and the left (resp. right, two-sided) ideals of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">: the correspondence is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\mapsto f(I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">↦</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I\mapsto f(I)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41d7a7ebbb92e0a053958ca3a2f4fdd8f0d3f3be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.045ex; height:2.843ex;" alt="{\displaystyle I\mapsto f(I)}"> and the pre-image ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J\mapsto f^{-1}(J)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo stretchy="false">↦</mo> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>J</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J\mapsto f^{-1}(J)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f007f013a7c92f84c287d9ba74e3d97dfdb58afc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.019ex; height:3.176ex;" alt="{\displaystyle J\mapsto f^{-1}(J)}">⁠. Moreover, for commutative rings, this bijective correspondence restricts to prime ideals, maximal ideals, and radical ideals (see the <a href="#Types_of_ideals">Types of ideals</a> section for the definitions of these ideals).</li> <li>(For those who know modules) If M is a left R-module and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subset M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>⊂</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S\subset M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71da27ce79329082f6bc54d5630d26814ec99a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.04ex; height:2.176ex;" alt="{\displaystyle S\subset M}"> a subset, then the <a href="/wiki/Annihilator_(ring_theory)" title="Annihilator (ring theory)">annihilator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ann} _{R}(S)=\{r\in R\mid rs=0,s\in S\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Ann</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>r</mi> <mo>∈</mo> <mi>R</mi> <mo>∣</mo> <mi>r</mi> <mi>s</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>s</mi> <mo>∈</mo> <mi>S</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Ann} _{R}(S)=\{r\in R\mid rs=0,s\in S\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b48f9d41cc21455d4896a650bb85a34ca1871d33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.994ex; height:2.843ex;" alt="{\displaystyle \operatorname {Ann} _{R}(S)=\{r\in R\mid rs=0,s\in S\}}"> of S is a left ideal. Given ideals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}},{\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}},{\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84b7742bed6d8d23c026def725678bf71ab31f99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {a}},{\mathfrak {b}}}"> of a commutative ring R, the R-annihilator of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathfrak {b}}+{\mathfrak {a}})/{\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathfrak {b}}+{\mathfrak {a}})/{\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a132e1b4e825a1195f6e24ee1ba3b7b6d36b161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.33ex; height:2.843ex;" alt="{\displaystyle ({\mathfrak {b}}+{\mathfrak {a}})/{\mathfrak {a}}}"> is an ideal of R called the <a href="/wiki/Ideal_quotient" title="Ideal quotient">ideal quotient</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47431c0c547b479e2cb895f75e15aefdb16bfd80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.193ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {b}}}"> and is denoted by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathfrak {a}}:{\mathfrak {b}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathfrak {a}}:{\mathfrak {b}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06c29c80c0558dbad6d05bb9090b4e0f1e921e54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.102ex; height:2.843ex;" alt="{\displaystyle ({\mathfrak {a}}:{\mathfrak {b}})}">⁠; it is an instance of <a href="/wiki/Idealizer" title="Idealizer">idealizer</a> in commutative algebra.</li> <li>Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{i},i\in S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mi>i</mi> <mo>∈</mo> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{i},i\in S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/443acab9de5a652c1649f247d53447b7b3efdf24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.138ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {a}}_{i},i\in S}"> be an <a href="/wiki/Ascending_chain" class="mw-redirect" title="Ascending chain">ascending chain</a> of left ideals in a ring R; i.e., <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"> is a totally ordered set and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{i}\subset {\mathfrak {a}}_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⊂</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{i}\subset {\mathfrak {a}}_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78bf7652da2a57e66746813ebefc23ed16ce26f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.133ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {a}}_{i}\subset {\mathfrak {a}}_{j}}"> for each ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i<j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo><</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i<j}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e60ff2d1b23e30fb2979e8c1536da03493f943cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.859ex; height:2.509ex;" alt="{\displaystyle i<j}">⁠. Then the union <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \bigcup _{i\in S}{\mathfrak {a}}_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>⋃</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>∈</mo> <mi>S</mi> </mrow> </munder> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle \bigcup _{i\in S}{\mathfrak {a}}_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e7c0431fbdf79ba4e0120f4c968a93515855053" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.241ex; height:3.009ex;" alt="{\displaystyle \textstyle \bigcup _{i\in S}{\mathfrak {a}}_{i}}"> is a left ideal of R. (Note: this fact remains true even if R is without the unity 1.)</li> <li>The above fact together with <a href="/wiki/Zorn%27s_lemma" title="Zorn's lemma">Zorn's lemma</a> proves the following: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\subset R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>⊂</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\subset R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d9a0d58ce4606a6c044315305acdd78e5c12cc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.638ex; height:2.176ex;" alt="{\displaystyle E\subset R}"> is a possibly empty subset and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{0}\subset R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>⊂</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{0}\subset R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca64b6fb3ed7a664f017410137b97c4165a457b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.079ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {a}}_{0}\subset R}"> is a left ideal that is disjoint from E, then there is an ideal that is maximal among the ideals containing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db87213ff7a2ba6fc50dba0e6bd4f97a11e45f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.217ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {a}}_{0}}"> and disjoint from E. (Again this is still valid if the ring R lacks the unity 1.) When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R\neq 0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6668d74ef5db38b2f2b6202160649ac2cd67e4e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.676ex;" alt="{\displaystyle R\neq 0}">, taking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{0}=(0)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{0}=(0)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fed4612e703a06dd903a32d18cbf6a212e7cec5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.287ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {a}}_{0}=(0)}"> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\{1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\{1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e322fda7d5d20b8cf69c1ba91cb67c692f234f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.361ex; height:2.843ex;" alt="{\displaystyle E=\{1\}}">⁠, in particular, there exists a left ideal that is maximal among proper left ideals (often simply called a maximal left ideal); see <a href="/wiki/Krull%27s_theorem" title="Krull's theorem">Krull's theorem</a> for more.</li> <li>An arbitrary union of ideals need not be an ideal, but the following is still true: given a possibly empty subset X of R, there is the smallest left ideal containing X, called the left ideal generated by X and is denoted by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle RX}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RX}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/048569f7cdffbaa866c54adec2003f700412270e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.744ex; height:2.176ex;" alt="{\displaystyle RX}">⁠. Such an ideal exists since it is the intersection of all left ideals containing X. Equivalently, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle RX}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RX}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/048569f7cdffbaa866c54adec2003f700412270e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.744ex; height:2.176ex;" alt="{\displaystyle RX}"> is the set of all the <a href="/wiki/Linear_combinations" class="mw-redirect" title="Linear combinations">(finite) left R-linear combinations</a> of elements of X over R: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle RX=\{r_{1}x_{1}+\dots +r_{n}x_{n}\mid n\in \mathbb {N} ,r_{i}\in R,x_{i}\in X\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>X</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>R</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>X</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RX=\{r_{1}x_{1}+\dots +r_{n}x_{n}\mid n\in \mathbb {N} ,r_{i}\in R,x_{i}\in X\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6870e2fd0ead5f6b2471373d621f92b9b7f87c49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.842ex; height:2.843ex;" alt="{\displaystyle RX=\{r_{1}x_{1}+\dots +r_{n}x_{n}\mid n\in \mathbb {N} ,r_{i}\in R,x_{i}\in X\}.}"></dd></dl></li></ul> <dl><dd>(since such a span is the smallest left ideal containing X.)<a href="#cite_note-13">[note 2]</a> A right (resp. two-sided) ideal generated by X is defined in the similar way. For "two-sided", one has to use linear combinations from both sides; i.e., <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle RXR=\{r_{1}x_{1}s_{1}+\dots +r_{n}x_{n}s_{n}\mid n\in \mathbb {N} ,r_{i}\in R,s_{i}\in R,x_{i}\in X\}.\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>X</mi> <mi>R</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>R</mi> <mo>,</mo> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>R</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>X</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RXR=\{r_{1}x_{1}s_{1}+\dots +r_{n}x_{n}s_{n}\mid n\in \mathbb {N} ,r_{i}\in R,s_{i}\in R,x_{i}\in X\}.\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4518c93bd11a3ae343de84983cc5e92b045bc9df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:64.975ex; height:2.843ex;" alt="{\displaystyle RXR=\{r_{1}x_{1}s_{1}+\dots +r_{n}x_{n}s_{n}\mid n\in \mathbb {N} ,r_{i}\in R,s_{i}\in R,x_{i}\in X\}.\,}"></dd></dl></dd></dl> <ul><li>A left (resp. right, two-sided) ideal generated by a single element x is called the principal left (resp. right, two-sided) ideal generated by x and is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Rx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Rx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8072625515d9f43b2fd354deb2126199df9bb4cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.094ex; height:2.176ex;" alt="{\displaystyle Rx}"> (resp. ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle xR,RxR}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mi>R</mi> <mo>,</mo> <mi>R</mi> <mi>x</mi> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle xR,RxR}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/072b30abbdff02e3abb77323731d18f5ae92ccc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.985ex; height:2.509ex;" alt="{\displaystyle xR,RxR}">⁠). The principal two-sided ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle RxR}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>x</mi> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RxR}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f995f1ea8aacb2b140aed2ad80b0b0cd62b2e53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.858ex; height:2.176ex;" alt="{\displaystyle RxR}"> is often also denoted by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd1f6d437f1742bfd5ffbdccd2477c07e9909e04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.139ex; height:2.843ex;" alt="{\displaystyle (x)}">⁠. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\{x_{1},\dots ,x_{n}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\{x_{1},\dots ,x_{n}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/100345e63f9659382d5a93de3bb0f87719a4ab07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.514ex; height:2.843ex;" alt="{\displaystyle X=\{x_{1},\dots ,x_{n}\}}"> is a finite set, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle RXR}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>X</mi> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RXR}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc2dab66d16ee3f4ca624b0ffc501888d806a204" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.508ex; height:2.176ex;" alt="{\displaystyle RXR}"> is also written as ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},\dots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (x_{1},\dots ,x_{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56204ada1ecf9dd5c6beddfdfd0f341cd69ff632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.92ex; height:2.843ex;" alt="{\displaystyle (x_{1},\dots ,x_{n})}">⁠.</li> <li>There is a bijective correspondence between ideals and <a href="/wiki/Congruence_relation" title="Congruence relation">congruence relations</a> (equivalence relations that respect the ring structure) on the ring: Given an ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> of a ring ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">⁠, let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\sim y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∼</mo> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\sim y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbd1014d850b7c883eb76301dd58c643e3c7e4eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.584ex; height:2.009ex;" alt="{\displaystyle x\sim y}"> if ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x-y\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x-y\in I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a38120b4e5728ed34204ea954872e3dcf828749" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.338ex; height:2.509ex;" alt="{\displaystyle x-y\in I}">⁠. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"> is a congruence relation on ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">⁠. Conversely, given a congruence relation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sim }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∼</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sim }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afcc42adfcfdc24d5c4c474869e5d8eaa78d1173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.307ex; margin-bottom: -0.478ex; width:1.808ex; height:1.343ex;" alt="{\displaystyle \sim }"> on ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">⁠, let ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=\{x\in R:x\sim 0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>R</mi> <mo>:</mo> <mi>x</mi> <mo>∼</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\{x\in R:x\sim 0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd6b44708ca3eb2e71fd3b2bd725d785b43b3fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.057ex; height:2.843ex;" alt="{\displaystyle I=\{x\in R:x\sim 0\}}">⁠. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is an ideal of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">⁠.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Types_of_ideals">Types of ideals</h2>[<a href="/w/index.php?title=Ideal_(ring_theory)&action=edit&section=5" title="Edit section: Types of ideals">edit</a>]</div> To simplify the description all rings are assumed to be commutative. The non-commutative case is discussed in detail in the respective articles. Ideals are important because they appear as kernels of ring homomorphisms and allow one to define <a href="/wiki/Factor_ring" class="mw-redirect" title="Factor ring">factor rings</a>. Different types of ideals are studied because they can be used to construct different types of factor rings. <ul><li><a href="/wiki/Maximal_ideal" title="Maximal ideal">Maximal ideal</a>: A proper ideal I is called a maximal ideal if there exists no other proper ideal J with I a proper subset of J. The factor ring of a maximal ideal is a <a href="/wiki/Simple_ring" title="Simple ring">simple ring</a> in general and is a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a> for commutative rings.<a href="#cite_note-14">[12]</a></li> <li><a href="/wiki/Minimal_ideal" title="Minimal ideal">Minimal ideal</a>: A nonzero ideal is called minimal if it contains no other nonzero ideal.</li> <li>Zero ideal: the ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ff0df9ef65c0572eb676580ce1c02b8ec40f694" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}">.<a href="#cite_note-15">[13]</a></li> <li>Unit ideal: the whole ring (being the ideal generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">).<a href="#cite_note-FOOTNOTEDummitFoote2004243-10">[9]</a></li> <li><a href="/wiki/Prime_ideal" title="Prime ideal">Prime ideal</a>: A proper ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is called a prime ideal if for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">⁠, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ab}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ab}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49337c5cf256196e2292f7047cb5da68c24ca95d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.227ex; height:2.176ex;" alt="{\displaystyle ab}"> is in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}">⁠, then at least one of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}"> is in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}">⁠. The factor ring of a prime ideal is a <a href="/wiki/Prime_ring" title="Prime ring">prime ring</a> in general and is an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a> for commutative rings.<a href="#cite_note-FOOTNOTEDummitFoote2004255-16">[14]</a></li> <li><a href="/wiki/Radical_of_an_ideal" title="Radical of an ideal">Radical ideal</a> or <a href="/wiki/Semiprime_ideal" class="mw-redirect" title="Semiprime ideal">semiprime ideal</a>: A proper ideal I is called radical or semiprime if for any a in R, if an is in I for some n, then a is in I. The factor ring of a radical ideal is a <a href="/wiki/Semiprime_ring" title="Semiprime ring">semiprime ring</a> for general rings, and is a <a href="/wiki/Reduced_ring" title="Reduced ring">reduced ring</a> for commutative rings.</li> <li><a href="/wiki/Primary_ideal" title="Primary ideal">Primary ideal</a>: An ideal I is called a primary ideal if for all a and b in R, if ab is in I, then at least one of a and bn is in I for some <a href="/wiki/Natural_number" title="Natural number">natural number</a> n. Every prime ideal is primary, but not conversely. A semiprime primary ideal is prime.</li> <li><a href="/wiki/Principal_ideal" title="Principal ideal">Principal ideal</a>: An ideal generated by one element.<a href="#cite_note-FOOTNOTEDummitFoote2004251-17">[15]</a></li> <li>Finitely generated ideal: This type of ideal is <a href="/wiki/Finitely_generated_module" title="Finitely generated module">finitely generated</a> as a module.</li> <li><a href="/wiki/Primitive_ideal" title="Primitive ideal">Primitive ideal</a>: A left primitive ideal is the <a href="/wiki/Annihilator_(ring_theory)" title="Annihilator (ring theory)">annihilator</a> of a <a href="/wiki/Simple_module" title="Simple module">simple</a> left <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a>.</li> <li><a href="/wiki/Irreducible_ideal" title="Irreducible ideal">Irreducible ideal</a>: An ideal is said to be irreducible if it cannot be written as an intersection of ideals that properly contain it.</li> <li>Comaximal ideals: Two ideals I, J are said to be comaximal if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x+y=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x+y=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/232443130147e73c8333f0d5fd7be981223e2854" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.586ex; height:2.509ex;" alt="{\displaystyle x+y=1}"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dec8caa8f241cb38a5348d7937b538227ad32c48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.342ex; height:2.176ex;" alt="{\displaystyle x\in I}"> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y\in J}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>∈</mo> <mi>J</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y\in J}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eecbdcfb812f80b87c1deacaed375451bbcae9eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.468ex; height:2.509ex;" alt="{\displaystyle y\in J}">⁠.</li> <li><a href="/wiki/Regular_ideal" title="Regular ideal">Regular ideal</a>: This term has multiple uses. See the article for a list.</li> <li><a href="/wiki/Nil_ideal" title="Nil ideal">Nil ideal</a>: An ideal is a nil ideal if each of its elements is nilpotent.</li> <li><a href="/wiki/Nilpotent_ideal" title="Nilpotent ideal">Nilpotent ideal</a>: Some power of it is zero.</li> <li><a href="/wiki/Parameter_ideal" class="mw-redirect" title="Parameter ideal">Parameter ideal</a>: an ideal generated by a <a href="/wiki/System_of_parameters" title="System of parameters">system of parameters</a>.</li> <li><a href="/wiki/Perfect_ideal" title="Perfect ideal">Perfect ideal</a>: A proper ideal I in a Noetherian ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> is called a perfect ideal if its <a href="/wiki/Grade_(ring_theory)" title="Grade (ring theory)">grade</a> equals the <a href="/wiki/Projective_dimension" class="mw-redirect" title="Projective dimension">projective dimension</a> of the associated quotient ring,<a href="#cite_note-Matsumura1-18">[16]</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textrm {grade}}(I)={\textrm {proj}}\dim(R/I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>grade</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>proj</mtext> </mrow> </mrow> <mi>dim</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textrm {grade}}(I)={\textrm {proj}}\dim(R/I)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b778dbcd18ae7d84a838d95d9118c74e748a170" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.889ex; height:2.843ex;" alt="{\displaystyle {\textrm {grade}}(I)={\textrm {proj}}\dim(R/I)}">⁠. A perfect ideal is <a href="/wiki/Unmixed_ideal" class="mw-redirect" title="Unmixed ideal">unmixed</a>.</li> <li><a href="/wiki/Cohen%E2%80%93Macaulay_ring#unmixed" title="Cohen–Macaulay ring">Unmixed ideal</a>: A proper ideal I in a Noetherian ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> is called an unmixed ideal (in height) if the height of I is equal to the height of every <a href="/wiki/Associated_prime" title="Associated prime">associated prime</a> P of R/I. (This is stronger than saying that R/I is <a href="/wiki/Equidimensionality" title="Equidimensionality">equidimensional</a>. See also <a href="/wiki/Minimal_prime_ideal#Equidimensional_ring" title="Minimal prime ideal">equidimensional ring</a>.</li></ul> Two other important terms using "ideal" are not always ideals of their ring. See their respective articles for details: <ul><li><a href="/wiki/Fractional_ideal" title="Fractional ideal">Fractional ideal</a>: This is usually defined when R is a commutative domain with <a href="/wiki/Quotient_field" class="mw-redirect" title="Quotient field">quotient field</a> K. Despite their names, fractional ideals are R submodules of K with a special property. If the fractional ideal is contained entirely in R, then it is truly an ideal of R.</li> <li><a href="/wiki/Invertible_ideal" class="mw-redirect" title="Invertible ideal">Invertible ideal</a>: Usually an invertible ideal A is defined as a fractional ideal for which there is another fractional ideal B such that AB = BA = R. Some authors may also apply "invertible ideal" to ordinary ring ideals A and B with AB = BA = R in rings other than domains.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Ideal_operations">Ideal operations</h2>[<a href="/w/index.php?title=Ideal_(ring_theory)&action=edit&section=6" title="Edit section: Ideal operations">edit</a>]</div> The sum and product of ideals are defined as follows. For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}"> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47431c0c547b479e2cb895f75e15aefdb16bfd80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.193ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {b}}}">⁠, left (resp. right) ideals of a ring R, their sum is <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}:=\{a+b\mid a\in {\mathfrak {a}}{\mbox{ and }}b\in {\mathfrak {b}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>+</mo> <mi>b</mi> <mo>∣</mo> <mi>a</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> and </mtext> </mstyle> </mrow> <mi>b</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}:=\{a+b\mid a\in {\mathfrak {a}}{\mbox{ and }}b\in {\mathfrak {b}}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e669cf421b16a8e56b68f16e06688a46d94d9d88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.443ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}:=\{a+b\mid a\in {\mathfrak {a}}{\mbox{ and }}b\in {\mathfrak {b}}\}}">,</dd></dl> which is a left (resp. right) ideal, and, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}},{\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}},{\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84b7742bed6d8d23c026def725678bf71ab31f99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {a}},{\mathfrak {b}}}"> are two-sided, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}{\mathfrak {b}}:=\{a_{1}b_{1}+\dots +a_{n}b_{n}\mid a_{i}\in {\mathfrak {a}}{\mbox{ and }}b_{i}\in {\mathfrak {b}},i=1,2,\dots ,n;{\mbox{ for }}n=1,2,\dots \},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>:=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>∣</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> and </mtext> </mstyle> </mrow> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext> for </mtext> </mstyle> </mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}{\mathfrak {b}}:=\{a_{1}b_{1}+\dots +a_{n}b_{n}\mid a_{i}\in {\mathfrak {a}}{\mbox{ and }}b_{i}\in {\mathfrak {b}},i=1,2,\dots ,n;{\mbox{ for }}n=1,2,\dots \},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c401f6cea1d5f022247e2c0bf88bc166daea7961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:76.642ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {a}}{\mathfrak {b}}:=\{a_{1}b_{1}+\dots +a_{n}b_{n}\mid a_{i}\in {\mathfrak {a}}{\mbox{ and }}b_{i}\in {\mathfrak {b}},i=1,2,\dots ,n;{\mbox{ for }}n=1,2,\dots \},}"></dd></dl> i.e. the product is the ideal generated by all products of the form ab with a in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}"> and b in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47431c0c547b479e2cb895f75e15aefdb16bfd80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.193ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {b}}}">⁠. Note <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f71b2aff6bc8562cf1239121a1289664ce3486a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.196ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}}"> is the smallest left (resp. right) ideal containing both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47431c0c547b479e2cb895f75e15aefdb16bfd80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.193ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {b}}}"> (or the union ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}\cup {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>∪</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}\cup {\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7c067cea3e612007a1e143a2b45266387ca74da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.938ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {a}}\cup {\mathfrak {b}}}">⁠), while the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}{\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}{\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beae22b0a00a8f2356d4ef75ad88560846b218fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.355ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {a}}{\mathfrak {b}}}"> is contained in the intersection of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}"> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47431c0c547b479e2cb895f75e15aefdb16bfd80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.193ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {b}}}">⁠. The distributive law holds for two-sided ideals ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}},{\mathfrak {b}},{\mathfrak {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}},{\mathfrak {b}},{\mathfrak {c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38d831b756d34b550a80f49218c64687ecbb368e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.328ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {a}},{\mathfrak {b}},{\mathfrak {c}}}">⁠, <ul><li>⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}({\mathfrak {b}}+{\mathfrak {c}})={\mathfrak {a}}{\mathfrak {b}}+{\mathfrak {a}}{\mathfrak {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}({\mathfrak {b}}+{\mathfrak {c}})={\mathfrak {a}}{\mathfrak {b}}+{\mathfrak {a}}{\mathfrak {c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d3c6aa3107ca6c282f124a1790d2959ce59565" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.271ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {a}}({\mathfrak {b}}+{\mathfrak {c}})={\mathfrak {a}}{\mathfrak {b}}+{\mathfrak {a}}{\mathfrak {c}}}">⁠,</li> <li>⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\mathfrak {a}}+{\mathfrak {b}}){\mathfrak {c}}={\mathfrak {a}}{\mathfrak {c}}+{\mathfrak {b}}{\mathfrak {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ({\mathfrak {a}}+{\mathfrak {b}}){\mathfrak {c}}={\mathfrak {a}}{\mathfrak {c}}+{\mathfrak {b}}{\mathfrak {c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f5b58860b7864d3f84a19582ee5a4a526cc4c20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.013ex; height:2.843ex;" alt="{\displaystyle ({\mathfrak {a}}+{\mathfrak {b}}){\mathfrak {c}}={\mathfrak {a}}{\mathfrak {c}}+{\mathfrak {b}}{\mathfrak {c}}}">⁠.</li></ul> If a product is replaced by an intersection, a partial distributive law holds: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}\cap ({\mathfrak {b}}+{\mathfrak {c}})\supset {\mathfrak {a}}\cap {\mathfrak {b}}+{\mathfrak {a}}\cap {\mathfrak {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>∩</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>⊃</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}\cap ({\mathfrak {b}}+{\mathfrak {c}})\supset {\mathfrak {a}}\cap {\mathfrak {b}}+{\mathfrak {a}}\cap {\mathfrak {c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65cd8609db815923ec5e0c885bd5cef512e4a40b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.018ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {a}}\cap ({\mathfrak {b}}+{\mathfrak {c}})\supset {\mathfrak {a}}\cap {\mathfrak {b}}+{\mathfrak {a}}\cap {\mathfrak {c}}}"></dd></dl> where the equality holds if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}"> contains <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47431c0c547b479e2cb895f75e15aefdb16bfd80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.193ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {b}}}"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21924b960341255be18e538e51404718f29cbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {c}}}">. Remark: The sum and the intersection of ideals is again an ideal; with these two operations as join and meet, the set of all ideals of a given ring forms a <a href="/wiki/Complete_lattice" title="Complete lattice">complete</a> <a href="/wiki/Modular_lattice" title="Modular lattice">modular lattice</a>. The lattice is not, in general, a <a href="/wiki/Distributive_lattice" title="Distributive lattice">distributive lattice</a>. The three operations of intersection, sum (or join), and product make the set of ideals of a commutative ring into a <a href="/wiki/Quantale" title="Quantale">quantale</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}},{\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}},{\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84b7742bed6d8d23c026def725678bf71ab31f99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.389ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {a}},{\mathfrak {b}}}"> are ideals of a commutative ring R, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}\cap {\mathfrak {b}}={\mathfrak {a}}{\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}\cap {\mathfrak {b}}={\mathfrak {a}}{\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43f6853db7400534c0e286d00481683488469c98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.391ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {a}}\cap {\mathfrak {b}}={\mathfrak {a}}{\mathfrak {b}}}"> in the following two cases (at least) <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}=(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}=(1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7128f42d3b9cd1d981b28089d9ce8cd82305fad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.266ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}=(1)}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}"> is generated by elements that form a <a href="/wiki/Regular_sequence" title="Regular sequence">regular sequence</a> modulo ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47431c0c547b479e2cb895f75e15aefdb16bfd80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.193ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {b}}}">⁠.</li></ul> (More generally, the difference between a product and an intersection of ideals is measured by the <a href="/wiki/Tor_functor" title="Tor functor">Tor functor</a>: ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Tor} _{1}^{R}(R/{\mathfrak {a}},R/{\mathfrak {b}})=({\mathfrak {a}}\cap {\mathfrak {b}})/{\mathfrak {a}}{\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>Tor</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>R</mi> </mrow> </msubsup> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>,</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Tor} _{1}^{R}(R/{\mathfrak {a}},R/{\mathfrak {b}})=({\mathfrak {a}}\cap {\mathfrak {b}})/{\mathfrak {a}}{\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bea766daa458ba97ea57b4d2f6c2924cfd1649a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.646ex; height:3.176ex;" alt="{\displaystyle \operatorname {Tor} _{1}^{R}(R/{\mathfrak {a}},R/{\mathfrak {b}})=({\mathfrak {a}}\cap {\mathfrak {b}})/{\mathfrak {a}}{\mathfrak {b}}}">⁠.<a href="#cite_note-19">[17]</a>) An integral domain is called a <a href="/wiki/Dedekind_domain" title="Dedekind domain">Dedekind domain</a> if for each pair of ideals <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}\subset {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>⊂</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}\subset {\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8185e98deffa597a93568b267ea4a8ec1b592f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.454ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {a}}\subset {\mathfrak {b}}}">, there is an ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b21924b960341255be18e538e51404718f29cbc0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.905ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {c}}}"> such that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {\mathfrak {a}}}={\mathfrak {b}}{\mathfrak {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {\mathfrak {a}}}={\mathfrak {b}}{\mathfrak {c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56c02ffd31632b80fb1562f87cf5d3f861f0b352" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.358ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {\mathfrak {a}}}={\mathfrak {b}}{\mathfrak {c}}}">⁠.<a href="#cite_note-FOOTNOTEMilnor19719-20">[18]</a> It can then be shown that every nonzero ideal of a Dedekind domain can be uniquely written as a product of maximal ideals, a generalization of the <a href="/wiki/Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>. <div class="mw-heading mw-heading2"><h2 id="Examples_of_ideal_operations">Examples of ideal operations</h2>[<a href="/w/index.php?title=Ideal_(ring_theory)&action=edit&section=7" title="Edit section: Examples of ideal operations">edit</a>]</div> In <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"> we have <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n)\cap (m)=\operatorname {lcm} (n,m)\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>lcm</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n)\cap (m)=\operatorname {lcm} (n,m)\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41df782de50276e8710951f6322e0a8497d85fed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.179ex; height:2.843ex;" alt="{\displaystyle (n)\cap (m)=\operatorname {lcm} (n,m)\mathbb {Z} }"></dd></dl> since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n)\cap (m)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>∩</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (n)\cap (m)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0df65bd1659e737379431aa403caf2f533eda828" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.636ex; height:2.843ex;" alt="{\displaystyle (n)\cap (m)}"> is the set of integers that are divisible by both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}">⁠. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\mathbb {C} [x,y,z,w]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\mathbb {C} [x,y,z,w]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/374e83752e18b61098f8285791f9195519f1ba21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.173ex; height:2.843ex;" alt="{\displaystyle R=\mathbb {C} [x,y,z,w]}"> and let ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}=(z,w),{\mathfrak {b}}=(x+z,y+w),{\mathfrak {c}}=(x+z,w)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}=(z,w),{\mathfrak {b}}=(x+z,y+w),{\mathfrak {c}}=(x+z,w)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e22bb012983ad26ae6fc3ac348bf8ac1ccaa61f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.746ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {a}}=(z,w),{\mathfrak {b}}=(x+z,y+w),{\mathfrak {c}}=(x+z,w)}">⁠. Then, <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}=(z,w,x+z,y+w)=(x,y,z,w)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo>,</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo>,</mo> <mi>y</mi> <mo>+</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}=(z,w,x+z,y+w)=(x,y,z,w)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b74ba042651905a0203dbf18223e4ae1e6722e5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.123ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {a}}+{\mathfrak {b}}=(z,w,x+z,y+w)=(x,y,z,w)}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}+{\mathfrak {c}}=(z,w,x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}+{\mathfrak {c}}=(z,w,x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87506f1b102dd09fe57f70a49549497566f26f59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.965ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {a}}+{\mathfrak {c}}=(z,w,x)}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}{\mathfrak {b}}=(z(x+z),z(y+w),w(x+z),w(y+w))=(z^{2}+xz,zy+wz,wx+wz,wy+w^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>z</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>w</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mi>w</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>+</mo> <mi>w</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>x</mi> <mi>z</mi> <mo>,</mo> <mi>z</mi> <mi>y</mi> <mo>+</mo> <mi>w</mi> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mi>x</mi> <mo>+</mo> <mi>w</mi> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mi>y</mi> <mo>+</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}{\mathfrak {b}}=(z(x+z),z(y+w),w(x+z),w(y+w))=(z^{2}+xz,zy+wz,wx+wz,wy+w^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59b7bcbb54bd40b7ed438203d08604c30a32c868" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:85.156ex; height:3.176ex;" alt="{\displaystyle {\mathfrak {a}}{\mathfrak {b}}=(z(x+z),z(y+w),w(x+z),w(y+w))=(z^{2}+xz,zy+wz,wx+wz,wy+w^{2})}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}{\mathfrak {c}}=(xz+z^{2},zw,xw+zw,w^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mi>z</mi> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> <mi>z</mi> <mi>w</mi> <mo>,</mo> <mi>x</mi> <mi>w</mi> <mo>+</mo> <mi>z</mi> <mi>w</mi> <mo>,</mo> <msup> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}{\mathfrak {c}}=(xz+z^{2},zw,xw+zw,w^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4202bf9016481bce06846956771fac4c2a42bcb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.537ex; height:3.176ex;" alt="{\displaystyle {\mathfrak {a}}{\mathfrak {c}}=(xz+z^{2},zw,xw+zw,w^{2})}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}\cap {\mathfrak {b}}={\mathfrak {a}}{\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}\cap {\mathfrak {b}}={\mathfrak {a}}{\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43f6853db7400534c0e286d00481683488469c98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.391ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {a}}\cap {\mathfrak {b}}={\mathfrak {a}}{\mathfrak {b}}}"> while <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}\cap {\mathfrak {c}}=(w,xz+z^{2})\neq {\mathfrak {a}}{\mathfrak {c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>∩</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo>,</mo> <mi>x</mi> <mi>z</mi> <mo>+</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>≠</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">c</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}\cap {\mathfrak {c}}=(w,xz+z^{2})\neq {\mathfrak {a}}{\mathfrak {c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a1e3d39a84273aa73fb6fa362dfe19f8e120ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.824ex; height:3.176ex;" alt="{\displaystyle {\mathfrak {a}}\cap {\mathfrak {c}}=(w,xz+z^{2})\neq {\mathfrak {a}}{\mathfrak {c}}}"></li></ul> In the first computation, we see the general pattern for taking the sum of two finitely generated ideals, it is the ideal generated by the union of their generators. In the last three we observe that products and intersections agree whenever the two ideals intersect in the zero ideal. These computations can be checked using <a href="/wiki/Macaulay_2" class="mw-redirect" title="Macaulay 2">Macaulay2</a>.<a href="#cite_note-21">[19]</a><a href="#cite_note-22">[20]</a><a href="#cite_note-23">[21]</a> <div class="mw-heading mw-heading2"><h2 id="Radical_of_a_ring">Radical of a ring</h2>[<a href="/w/index.php?title=Ideal_(ring_theory)&action=edit&section=8" title="Edit section: Radical of a ring">edit</a>]</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/Radical_of_a_ring" title="Radical of a ring">Radical of a ring</a></div> Ideals appear naturally in the study of modules, especially in the form of a radical. <dl><dd>For simplicity, we work with commutative rings but, with some changes, the results are also true for non-commutative rings.</dd></dl> Let R be a commutative ring. By definition, a <a href="/wiki/Primitive_ideal" title="Primitive ideal">primitive ideal</a> of R is the annihilator of a (nonzero) <a href="/wiki/Simple_module" title="Simple module">simple R-module</a>. The <a href="/wiki/Jacobson_radical" title="Jacobson radical">Jacobson radical</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=\operatorname {Jac} (R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mi>Jac</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=\operatorname {Jac} (R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f517c268f684e8c41dbae81852e9483886d9b52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.533ex; height:2.843ex;" alt="{\displaystyle J=\operatorname {Jac} (R)}"> of R is the intersection of all primitive ideals. Equivalently, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=\bigcap _{{\mathfrak {m}}{\text{ maximal ideals}}}{\mathfrak {m}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> maximal ideals</mtext> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=\bigcap _{{\mathfrak {m}}{\text{ maximal ideals}}}{\mathfrak {m}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76512802934a4fea2531831a5df345edc2077630" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.001ex; height:5.676ex;" alt="{\displaystyle J=\bigcap _{{\mathfrak {m}}{\text{ maximal ideals}}}{\mathfrak {m}}.}"></dd></dl> Indeed, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"> is a simple module and x is a nonzero element in M, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Rx=M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>x</mi> <mo>=</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Rx=M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3609669182bce8c4a40e3e5d26b4c522bf6413f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.634ex; height:2.176ex;" alt="{\displaystyle Rx=M}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R/\operatorname {Ann} (M)=R/\operatorname {Ann} (x)\simeq M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>Ann</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>Ann</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>≃</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R/\operatorname {Ann} (M)=R/\operatorname {Ann} (x)\simeq M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fe62d9e34ad2b758483098a7966649f4a80feab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.313ex; height:2.843ex;" alt="{\displaystyle R/\operatorname {Ann} (M)=R/\operatorname {Ann} (x)\simeq M}">, meaning <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ann} (M)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Ann</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Ann} (M)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d5fca7f3f63b96e5a5731e321ff72ac10831faa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.58ex; height:2.843ex;" alt="{\displaystyle \operatorname {Ann} (M)}"> is a maximal ideal. Conversely, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0e9162e96758157a34a6e44967288b481a7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}"> is a maximal ideal, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0e9162e96758157a34a6e44967288b481a7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}"> is the annihilator of the simple R-module ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R/{\mathfrak {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R/{\mathfrak {m}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/695788b852be9fd463c7e135db432c98add7f7a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.709ex; height:2.843ex;" alt="{\displaystyle R/{\mathfrak {m}}}">⁠. There is also another characterization (the proof is not hard): <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=\{x\in R\mid 1-yx\,{\text{ is a unit element for every }}y\in R\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∈</mo> <mi>R</mi> <mo>∣</mo> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> is a unit element for every </mtext> </mrow> <mi>y</mi> <mo>∈</mo> <mi>R</mi> <mo fence="false" stretchy="false">}</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=\{x\in R\mid 1-yx\,{\text{ is a unit element for every }}y\in R\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac13c2d667626008dab4c52540b1f4b2f037744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.069ex; height:2.843ex;" alt="{\displaystyle J=\{x\in R\mid 1-yx\,{\text{ is a unit element for every }}y\in R\}.}"></dd></dl> For a not-necessarily-commutative ring, it is a general fact that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-yx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-yx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e6722b7a5782dacce05ce7dd2a562c7d7b87f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.488ex; height:2.509ex;" alt="{\displaystyle 1-yx}"> is a <a href="/wiki/Unit_element" class="mw-redirect" title="Unit element">unit element</a> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-xy}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>−</mo> <mi>x</mi> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1-xy}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f99d8184d37b26d9f928168af1a54e5b2e31a39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.488ex; height:2.509ex;" alt="{\displaystyle 1-xy}"> is (see the link) and so this last characterization shows that the radical can be defined both in terms of left and right primitive ideals. The following simple but important fact (<a href="/wiki/Nakayama%27s_lemma" title="Nakayama's lemma">Nakayama's lemma</a>) is built-in to the definition of a Jacobson radical: if M is a module such that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle JM=M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mi>M</mi> <mo>=</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle JM=M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f853bb23ba516c64ef5ba13d6de4c7ed65ff47c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.454ex; height:2.176ex;" alt="{\displaystyle JM=M}">⁠, then M does not admit a <a href="/wiki/Maximal_submodule" class="mw-redirect" title="Maximal submodule">maximal submodule</a>, since if there is a maximal submodule ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\subsetneq M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>⊊</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\subsetneq M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e74671eab7e1a0fa6e3fb3b15987d63a7377aa4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.124ex; height:2.676ex;" alt="{\displaystyle L\subsetneq M}">⁠, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J\cdot (M/L)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>L</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J\cdot (M/L)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6fdc6b117f42fb697583bb6993ed89e707c1d96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.408ex; height:2.843ex;" alt="{\displaystyle J\cdot (M/L)=0}"> and so ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=JM\subset L\subsetneq M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mi>J</mi> <mi>M</mi> <mo>⊂</mo> <mi>L</mi> <mo>⊊</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=JM\subset L\subsetneq M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1fc161ce606743784e9ecedf0b1f155adc1d657" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.676ex; height:2.676ex;" alt="{\displaystyle M=JM\subset L\subsetneq M}">⁠, a contradiction. Since a nonzero <a href="/wiki/Finitely_generated_module" title="Finitely generated module">finitely generated module</a> admits a maximal submodule, in particular, one has: <dl><dd>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle JM=M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mi>M</mi> <mo>=</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle JM=M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f853bb23ba516c64ef5ba13d6de4c7ed65ff47c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.454ex; height:2.176ex;" alt="{\displaystyle JM=M}"> and M is finitely generated, then ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b4002e03449266e23b8766bed86d22d1b1c3067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.703ex; height:2.176ex;" alt="{\displaystyle M=0}">⁠.</dd></dl> A maximal ideal is a prime ideal and so one has <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {nil} (R)=\bigcap _{{\mathfrak {p}}{\text{ prime ideals }}}{\mathfrak {p}}\subset \operatorname {Jac} (R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>nil</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>⋂</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext> prime ideals </mtext> </mrow> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo>⊂</mo> <mi>Jac</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {nil} (R)=\bigcap _{{\mathfrak {p}}{\text{ prime ideals }}}{\mathfrak {p}}\subset \operatorname {Jac} (R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f4eb2b81c946121550c41ae0774800c984f6fff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:31.576ex; height:5.843ex;" alt="{\displaystyle \operatorname {nil} (R)=\bigcap _{{\mathfrak {p}}{\text{ prime ideals }}}{\mathfrak {p}}\subset \operatorname {Jac} (R)}"></dd></dl> where the intersection on the left is called the <a href="/wiki/Nilradical_of_a_ring" title="Nilradical of a ring">nilradical</a> of R. As it turns out, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {nil} (R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>nil</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {nil} (R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f5a209bd2b972d67a525ec406e4ec50e456a0b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.16ex; height:2.843ex;" alt="{\displaystyle \operatorname {nil} (R)}"> is also the set of <a href="/wiki/Nilpotent_element" class="mw-redirect" title="Nilpotent element">nilpotent elements</a> of R. If R is an <a href="/wiki/Artinian_ring" title="Artinian ring">Artinian ring</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Jac} (R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Jac</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Jac} (R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fff3146e8831cf50a4ccbc11428c1fedef9a5ff8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.963ex; height:2.843ex;" alt="{\displaystyle \operatorname {Jac} (R)}"> is nilpotent and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {nil} (R)=\operatorname {Jac} (R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>nil</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>Jac</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {nil} (R)=\operatorname {Jac} (R)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3401d9608d0567d97c2b7d6f6276565d559d420" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.221ex; height:2.843ex;" alt="{\displaystyle \operatorname {nil} (R)=\operatorname {Jac} (R)}">⁠. (Proof: first note the DCC implies <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{n}=J^{n+1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{n}=J^{n+1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2f89a9c2cb450b8a62b0c2dfca1835999d88721" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.688ex; height:2.676ex;" alt="{\displaystyle J^{n}=J^{n+1}}"> for some n. If (DCC) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}\supsetneq \operatorname {Ann} (J^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>⊋</mo> <mi>Ann</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}\supsetneq \operatorname {Ann} (J^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8df32658a2d0b3e72842a18fa5376add5cdc981c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.143ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {a}}\supsetneq \operatorname {Ann} (J^{n})}"> is an ideal properly minimal over the latter, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J\cdot ({\mathfrak {a}}/\operatorname {Ann} (J^{n}))=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>Ann</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J\cdot ({\mathfrak {a}}/\operatorname {Ann} (J^{n}))=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0122264976ee2457800355562f46f23bb1590d1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.814ex; height:2.843ex;" alt="{\displaystyle J\cdot ({\mathfrak {a}}/\operatorname {Ann} (J^{n}))=0}">. That is, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J^{n}{\mathfrak {a}}=J^{n+1}{\mathfrak {a}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>=</mo> <msup> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J^{n}{\mathfrak {a}}=J^{n+1}{\mathfrak {a}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a76f78637b0ea052044af4ef7f64b927e3b64c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.273ex; height:2.676ex;" alt="{\displaystyle J^{n}{\mathfrak {a}}=J^{n+1}{\mathfrak {a}}=0}">⁠, a contradiction.) <div class="mw-heading mw-heading2"><h2 id="Extension_and_contraction_of_an_ideal">Extension and contraction of an ideal</h2>[<a href="/w/index.php?title=Ideal_(ring_theory)&action=edit&section=9" title="Edit section: Extension and contraction of an ideal">edit</a>]</div> Let A and B be two <a href="/wiki/Commutative_ring" title="Commutative ring">commutative rings</a>, and let f : A → B be a <a href="/wiki/Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}"> is an ideal in A, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f({\mathfrak {a}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f({\mathfrak {a}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97afa190a4eff25d9f1f69552ed3e00da7401732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.25ex; height:2.843ex;" alt="{\displaystyle f({\mathfrak {a}})}"> need not be an ideal in B (e.g. take f to be the <a href="/wiki/Inclusion_map" title="Inclusion map">inclusion</a> of the ring of integers Z into the field of rationals Q). The extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}^{e}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/567fb2fbe6cbc450bf237481f07eafe5261db1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.161ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {a}}^{e}}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}"> in B is defined to be the ideal in B generated by ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f({\mathfrak {a}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f({\mathfrak {a}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97afa190a4eff25d9f1f69552ed3e00da7401732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.25ex; height:2.843ex;" alt="{\displaystyle f({\mathfrak {a}})}">⁠. Explicitly, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}^{e}={\Big \{}\sum y_{i}f(x_{i}):x_{i}\in {\mathfrak {a}},y_{i}\in B{\Big \}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">{</mo> </mrow> </mrow> <mo>∑</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>:</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∈</mo> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.623em" minsize="1.623em">}</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}^{e}={\Big \{}\sum y_{i}f(x_{i}):x_{i}\in {\mathfrak {a}},y_{i}\in B{\Big \}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/600e7aff83606f579660ba987b72589fd6e2c543" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.292ex; height:4.843ex;" alt="{\displaystyle {\mathfrak {a}}^{e}={\Big \{}\sum y_{i}f(x_{i}):x_{i}\in {\mathfrak {a}},y_{i}\in B{\Big \}}}"></dd></dl> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47431c0c547b479e2cb895f75e15aefdb16bfd80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.193ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {b}}}"> is an ideal of B, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}({\mathfrak {b}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f^{-1}({\mathfrak {b}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a024f0af20b50482ef08ed0c3d15d73936b7fb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.655ex; height:3.176ex;" alt="{\displaystyle f^{-1}({\mathfrak {b}})}"> is always an ideal of A, called the contraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}^{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42a0b993972c26c8d86ad67b82da92c14d613c9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.137ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {b}}^{c}}"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47431c0c547b479e2cb895f75e15aefdb16bfd80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.193ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {b}}}"> to A. Assuming f : A → B is a ring homomorphism, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}"> is an ideal in A, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47431c0c547b479e2cb895f75e15aefdb16bfd80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.193ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {b}}}"> is an ideal in B, then: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47431c0c547b479e2cb895f75e15aefdb16bfd80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.193ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {b}}}"> is prime in B <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Rightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇒</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Rightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/469b737d167b9b28a74e27c7f5e35b5ea9256100" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Rightarrow }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}^{c}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}^{c}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42a0b993972c26c8d86ad67b82da92c14d613c9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.137ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {b}}^{c}}"> is prime in A.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}^{ec}\supseteq {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>c</mi> </mrow> </msup> <mo>⊇</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}^{ec}\supseteq {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/446deaf532a897d293336ce672090b6b91729177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.134ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {a}}^{ec}\supseteq {\mathfrak {a}}}"></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}^{ce}\subseteq {\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>e</mi> </mrow> </msup> <mo>⊆</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}^{ce}\subseteq {\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41119aed93a35f7b083af784cc0ab957c59006ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.194ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {b}}^{ce}\subseteq {\mathfrak {b}}}"></li></ul> It is false, in general, that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}"> being prime (or maximal) in A implies that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}^{e}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/567fb2fbe6cbc450bf237481f07eafe5261db1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.161ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {a}}^{e}}"> is prime (or maximal) in B. Many classic examples of this stem from algebraic number theory. For example, <a href="/wiki/Embedding" title="Embedding">embedding</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} \to \mathbb {Z} \left\lbrack i\right\rbrack }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow> <mo>[</mo> <mi>i</mi> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \to \mathbb {Z} \left\lbrack i\right\rbrack }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7a34f76b881963b253d3f7bdc3df992914de8a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.198ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} \to \mathbb {Z} \left\lbrack i\right\rbrack }">⁠. In <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=\mathbb {Z} \left\lbrack i\right\rbrack }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow> <mo>[</mo> <mi>i</mi> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=\mathbb {Z} \left\lbrack i\right\rbrack }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13c1527985ec7bc0afd350edf7e151d81e8f10f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.896ex; height:2.843ex;" alt="{\displaystyle B=\mathbb {Z} \left\lbrack i\right\rbrack }">, the element 2 factors as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2=(1+i)(1-i)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2=(1+i)(1-i)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3dbaee0376fb70c561f30c28c7d6417b8b916eef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.49ex; height:2.843ex;" alt="{\displaystyle 2=(1+i)(1-i)}"> where (one can show) neither of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1+i,1-i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo>,</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1+i,1-i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2578759e6868f7ae8dcf599a632cf502d09fd096" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.645ex; height:2.509ex;" alt="{\displaystyle 1+i,1-i}"> are units in B. So <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2)^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2)^{e}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0420b40e3536cf76a034943376ab6818b70212a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.97ex; height:2.843ex;" alt="{\displaystyle (2)^{e}}"> is not prime in B (and therefore not maximal, as well). Indeed, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1\pm i)^{2}=\pm 2i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>±</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>±</mo> <mn>2</mn> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1\pm i)^{2}=\pm 2i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62f0b708a3b5ad404b3864ad5b8d033e52e28e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.54ex; height:3.176ex;" alt="{\displaystyle (1\pm i)^{2}=\pm 2i}"> shows that ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1+i)=((1-i)-(1-i)^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+i)=((1-i)-(1-i)^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d400da5702b53ba32b77362f8e70684004064c83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.646ex; height:3.176ex;" alt="{\displaystyle (1+i)=((1-i)-(1-i)^{2})}">⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-i)=((1+i)-(1+i)^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1-i)=((1+i)-(1+i)^{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2618c2bf49497b7a32f537c15b8d8aa84e22471b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.646ex; height:3.176ex;" alt="{\displaystyle (1-i)=((1+i)-(1+i)^{2})}">⁠, and therefore ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2)^{e}=(1+i)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>i</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2)^{e}=(1+i)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/593ee579513b499408317c9130e6ccb92a9162cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.737ex; height:3.176ex;" alt="{\displaystyle (2)^{e}=(1+i)^{2}}">⁠. On the other hand, if f is <a href="/wiki/Surjective_function" title="Surjective function">surjective</a> and <a href="/wiki/Kernel_(algebra)" title="Kernel (algebra)"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}\supseteq \ker f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>⊇</mo> <mi>ker</mi> <mo>⁡</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}\supseteq \ker f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e5575d3fec9db772ef2d0a3cb4cd5fb8a4599ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.098ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {a}}\supseteq \ker f}"></a> then: <ul><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}^{ec}={\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>c</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}^{ec}={\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8238afae1089b0aef8b952b878fc46439dc73798" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.134ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {a}}^{ec}={\mathfrak {a}}}"> and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}^{ce}={\mathfrak {b}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> <mi>e</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">b</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}^{ce}={\mathfrak {b}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff29b70c37f4d19c47e927ba523b3eef8e0607c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.194ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {b}}^{ce}={\mathfrak {b}}}">⁠.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}"> is a <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a> in A <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}^{e}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/567fb2fbe6cbc450bf237481f07eafe5261db1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.161ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {a}}^{e}}"> is a prime ideal in B.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16f656feeddb5d98500bb4d3fc31038d0b87484b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {a}}}"> is a <a href="/wiki/Maximal_ideal" title="Maximal ideal">maximal ideal</a> in A <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Leftrightarrow }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">⇔</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Leftrightarrow }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64812e13399c20cf3ce94e049d3bb2d85f26abcf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.324ex; height:1.843ex;" alt="{\displaystyle \Leftrightarrow }"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}^{e}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/567fb2fbe6cbc450bf237481f07eafe5261db1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.161ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {a}}^{e}}"> is a maximal ideal in B.</li></ul> Remark: Let K be a <a href="/wiki/Field_extension" title="Field extension">field extension</a> of L, and let B and A be the <a href="/wiki/Ring_of_integers" title="Ring of integers">rings of integers</a> of K and L, respectively. Then B is an <a href="/wiki/Integral_extension" class="mw-redirect" title="Integral extension">integral extension</a> of A, and we let f be the <a href="/wiki/Inclusion_map" title="Inclusion map">inclusion map</a> from A to B. The behaviour of a <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}={\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">a</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}={\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffc1f22fab13d0a9090ba5b80489756700ad2152" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.423ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {a}}={\mathfrak {p}}}"> of A under extension is one of the central problems of <a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">algebraic number theory</a>. The following is sometimes useful:<a href="#cite_note-FOOTNOTEAtiyahMacdonald1969Proposition_3.16-24">[22]</a> a prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}"> is a contraction of a prime ideal if and only if ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}={\mathfrak {p}}^{ec}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> <mi>c</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}={\mathfrak {p}}^{ec}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7de2b7ea3a4b96816e98180296db2756800b6be2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.134ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {p}}={\mathfrak {p}}^{ec}}">⁠. (Proof: Assuming the latter, note <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}^{e}B_{\mathfrak {p}}=B_{\mathfrak {p}}\Rightarrow {\mathfrak {p}}^{e}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> <mo stretchy="false">⇒</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}^{e}B_{\mathfrak {p}}=B_{\mathfrak {p}}\Rightarrow {\mathfrak {p}}^{e}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e00d101cd7fd30869e4a1daf7f6e3f335a337afd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.671ex; height:3.009ex;" alt="{\displaystyle {\mathfrak {p}}^{e}B_{\mathfrak {p}}=B_{\mathfrak {p}}\Rightarrow {\mathfrak {p}}^{e}}"> intersects ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A-{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0424c5c918f46b425bec0323d7070aceb844ce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.746ex; height:2.509ex;" alt="{\displaystyle A-{\mathfrak {p}}}">⁠, a contradiction. Now, the prime ideals of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac80226ccaff807ed073f88562198cf6b2ee9140" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.818ex; height:2.843ex;" alt="{\displaystyle B_{\mathfrak {p}}}"> correspond to those in B that are disjoint from ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A-{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0424c5c918f46b425bec0323d7070aceb844ce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.746ex; height:2.509ex;" alt="{\displaystyle A-{\mathfrak {p}}}">⁠. Hence, there is a prime ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">q</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79b0448036d3f2e513dd80d94095c45d81510269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.137ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {q}}}"> of B, disjoint from ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A-{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0424c5c918f46b425bec0323d7070aceb844ce1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.746ex; height:2.509ex;" alt="{\displaystyle A-{\mathfrak {p}}}">⁠, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {q}}B_{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">q</mi> </mrow> </mrow> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {q}}B_{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/97ab88b3356a637eaf4ef810bd5edcbd8fd1bd34" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.955ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {q}}B_{\mathfrak {p}}}"> is a maximal ideal containing ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}^{e}B_{\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>e</mi> </mrow> </msup> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}^{e}B_{\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a16d6bf3ab8d7eebae6cde8ca13146499558ab9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.979ex; height:3.009ex;" alt="{\displaystyle {\mathfrak {p}}^{e}B_{\mathfrak {p}}}">⁠. One then checks that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {q}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">q</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {q}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79b0448036d3f2e513dd80d94095c45d81510269" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.137ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {q}}}"> lies over ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">p</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14c125cdf81ac25d76edc2e8d557302c9f555a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}">⁠. The converse is obvious.) <div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2>[<a href="/w/index.php?title=Ideal_(ring_theory)&action=edit&section=10" title="Edit section: Generalizations">edit</a>]</div> Ideals can be generalized to any <a href="/wiki/Monoid_object" class="mw-redirect" title="Monoid object">monoid object</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (R,\otimes )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mo>⊗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (R,\otimes )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/381351d85a4dd769ed4754bccd3ac154b33eb8eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.415ex; height:2.843ex;" alt="{\displaystyle (R,\otimes )}">⁠, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> is the object where the <a href="/wiki/Monoid" title="Monoid">monoid</a> structure has been <a href="/wiki/Forgetful_functor" title="Forgetful functor">forgotten</a>. A left ideal of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> is a <a href="/wiki/Subobject" title="Subobject">subobject</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> that "absorbs multiplication from the left by elements of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">⁠"; that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is a left ideal if it satisfies the following two conditions: <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is a <a href="/wiki/Subobject" title="Subobject">subobject</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></li> <li>For every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in (R,\otimes )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mo>⊗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in (R,\otimes )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5e74c2a65fedb097c537a19b79cb70dc06ec39d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.305ex; height:2.843ex;" alt="{\displaystyle r\in (R,\otimes )}"> and every ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in (I,\otimes )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo>,</mo> <mo>⊗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\in (I,\otimes )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/816ea985ba9ecda6d173fdd261ab659975103d02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.993ex; height:2.843ex;" alt="{\displaystyle x\in (I,\otimes )}">⁠, the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\otimes x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>⊗</mo> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\otimes x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a731dd8bbb934bba1816b3839cf98f1b9c71eade" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.219ex; height:2.176ex;" alt="{\displaystyle r\otimes x}"> is in ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (I,\otimes )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>I</mi> <mo>,</mo> <mo>⊗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (I,\otimes )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1f835400db0d2f824540073714410eff4094ad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.823ex; height:2.843ex;" alt="{\displaystyle (I,\otimes )}">⁠.</li></ol> A right ideal is defined with the condition "⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\otimes x\in (I,\otimes )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>⊗</mo> <mi>x</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo>,</mo> <mo>⊗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\otimes x\in (I,\otimes )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7aaf31fa10c6012158baafb9ca935dc040d4826" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.883ex; height:2.843ex;" alt="{\displaystyle r\otimes x\in (I,\otimes )}">⁠" replaced by "'⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\otimes r\in (I,\otimes )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⊗</mo> <mi>r</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mi>I</mi> <mo>,</mo> <mo>⊗</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\otimes r\in (I,\otimes )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db856490a7552de1ac480b6063d5991e48f9a33a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.883ex; height:2.843ex;" alt="{\displaystyle x\otimes r\in (I,\otimes )}">⁠". A two-sided ideal is a left ideal that is also a right ideal, and is sometimes simply called an ideal. When <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> is a commutative monoid object respectively, the definitions of left, right, and two-sided ideal coincide, and the term ideal is used alone. An ideal can also be thought of as a specific type of <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">R-module</a>. If we consider <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> as a left <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">-module (by left multiplication), then a left ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is really just a left <a href="/wiki/Module_(mathematics)#Submodules_and_homomorphisms" title="Module (mathematics)">sub-module</a> of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">⁠. In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is a left (right) ideal of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"> if and only if it is a left (right) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">-module that is a subset of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">⁠. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is a two-sided ideal if it is a sub-<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">-bimodule of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}">⁠. Example: If we let ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R=\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7d60fc180e76c47f663220cdb9ff8c6f564c0f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.413ex; height:2.176ex;" alt="{\displaystyle R=\mathbb {Z} }">⁠, an ideal of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"> is an abelian group that is a subset of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">⁠, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb660446d851fca07d1cfdd36a71c022ad617d23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.591ex; height:2.176ex;" alt="{\displaystyle m\mathbb {Z} }"> for some ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12e01e20e2bc24ab21cca6a82edddec5c1a20ec5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.431ex; height:2.176ex;" alt="{\displaystyle m\in \mathbb {Z} }">⁠. So these give all the ideals of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">⁠. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Ideal_(ring_theory)&action=edit&section=11" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Modular_arithmetic" title="Modular arithmetic">Modular arithmetic</a></li> <li><a href="/wiki/Noether_isomorphism_theorem" class="mw-redirect" title="Noether isomorphism theorem">Noether isomorphism theorem</a></li> <li><a href="/wiki/Boolean_prime_ideal_theorem" title="Boolean prime ideal theorem">Boolean prime ideal theorem</a></li> <li><a href="/wiki/Ideal_theory" title="Ideal theory">Ideal theory</a></li> <li><a href="/wiki/Ideal_(order_theory)" title="Ideal (order theory)">Ideal (order theory)</a></li> <li><a href="/wiki/Ideal_norm" title="Ideal norm">Ideal norm</a></li> <li><a href="/wiki/Splitting_of_prime_ideals_in_Galois_extensions" title="Splitting of prime ideals in Galois extensions">Splitting of prime ideals in Galois extensions</a></li> <li><a href="/wiki/Ideal_sheaf" title="Ideal sheaf">Ideal sheaf</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Ideal_(ring_theory)&action=edit&section=12" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-9"><a href="#cite_ref-9">^</a> Some authors call the zero and unit ideals of a ring R the trivial ideals of R. </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> If R does not have a unit, then the internal descriptions above must be modified slightly. In addition to the finite sums of products of things in X with things in R, we must allow the addition of n-fold sums of the form x + x + ... + x, and n-fold sums of the form (−x) + (−x) + ... + (−x) for every x in X and every n in the natural numbers. When R has a unit, this extra requirement becomes superfluous. </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Ideal_(ring_theory)&action=edit&section=13" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-Stillwell-1">^ <a href="#cite_ref-Stillwell_1-0">a</a> <a href="#cite_ref-Stillwell_1-1">b</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFJohn_Stillwell2010" class="citation book cs1">John Stillwell (2010). Mathematics and its history. p. 439.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+and+its+history&rft.pages=439&rft.date=2010&rft.au=John+Stillwell&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"> </li> <li id="cite_note-flt-2"><a href="#cite_ref-flt_2-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHarold_M._Edwards1977" class="citation book cs1">Harold M. Edwards (1977). Fermat's last theorem. A genetic introduction to algebraic number theory. p. 76.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fermat%27s+last+theorem.+A+genetic+introduction+to+algebraic+number+theory&rft.pages=76&rft.date=1977&rft.au=Harold+M.+Edwards&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"> </li> <li id="cite_note-everest_ward-3"><a href="#cite_ref-everest_ward_3-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEverest_G.,_Ward_T.2005" class="citation book cs1">Everest G., Ward T. (2005). An introduction to number theory. p. 83.</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+number+theory&rft.pages=83&rft.date=2005&rft.au=Everest+G.%2C+Ward+T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFDummitFoote2004">Dummit & Foote 2004</a>, p. 242 </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFDummitFoote2004">Dummit & Foote 2004</a>, § 10.1., Examples (1). </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFDummitFoote2004">Dummit & Foote 2004</a>, § 10.1., Proposition 3. </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <a href="#CITEREFDummitFoote2004">Dummit & Foote 2004</a>, Ch. 7, Proposition 6. </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> <a href="#CITEREFDummitFoote2004">Dummit & Foote 2004</a>, Ch. 7, Theorem 7. </li> <li id="cite_note-FOOTNOTEDummitFoote2004243-10">^ <a href="#cite_ref-FOOTNOTEDummitFoote2004243_10-0">a</a> <a href="#cite_ref-FOOTNOTEDummitFoote2004243_10-1">b</a> <a href="#cite_ref-FOOTNOTEDummitFoote2004243_10-2">c</a> <a href="#cite_ref-FOOTNOTEDummitFoote2004243_10-3">d</a> <a href="#CITEREFDummitFoote2004">Dummit & Foote (2004)</a>, p. 243. </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <a href="#CITEREFLang2005">Lang 2005</a>, Section III.2 </li> <li id="cite_note-FOOTNOTEDummitFoote2004244-12"><a href="#cite_ref-FOOTNOTEDummitFoote2004244_12-0">^</a> <a href="#CITEREFDummitFoote2004">Dummit & Foote (2004)</a>, p. 244. </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> Because simple commutative rings are fields. See <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLam2001" class="citation book cs1">Lam (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=f15FyZuZ3-4C&pg=PA39&dq=%22simple+commutative+rings%22">A First Course in Noncommutative Rings</a>. p. 39.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+First+Course+in+Noncommutative+Rings&rft.pages=39&rft.date=2001&rft.au=Lam&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Df15FyZuZ3-4C%26pg%3DPA39%26dq%3D%2522simple%2Bcommutative%2Brings%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/ZeroIdeal.html">"Zero ideal"</a>. Math World. 22 Aug 2024.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Math+World&rft.atitle=Zero+ideal&rft.date=2024-08-22&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FZeroIdeal.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"> </li> <li id="cite_note-FOOTNOTEDummitFoote2004255-16"><a href="#cite_ref-FOOTNOTEDummitFoote2004255_16-0">^</a> <a href="#CITEREFDummitFoote2004">Dummit & Foote (2004)</a>, p. 255. </li> <li id="cite_note-FOOTNOTEDummitFoote2004251-17"><a href="#cite_ref-FOOTNOTEDummitFoote2004251_17-0">^</a> <a href="#CITEREFDummitFoote2004">Dummit & Foote (2004)</a>, p. 251. </li> <li id="cite_note-Matsumura1-18"><a href="#cite_ref-Matsumura1_18-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMatsumura1987" class="citation book cs1"><a href="/wiki/Hideyuki_Matsumura" title="Hideyuki Matsumura">Matsumura, Hideyuki</a> (1987). <a rel="nofollow" class="external text" href="https://www.cambridge.org/core/books/commutative-ring-theory/02819830750568B06C16E6199F3562C1">Commutative Ring Theory</a>. Cambridge: Cambridge University Press. p. 132. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781139171762" title="Special:BookSources/9781139171762"><bdi>9781139171762</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.place=Cambridge&rft.pages=132&rft.pub=Cambridge+University+Press&rft.date=1987&rft.isbn=9781139171762&rft.aulast=Matsumura&rft.aufirst=Hideyuki&rft_id=https%3A%2F%2Fwww.cambridge.org%2Fcore%2Fbooks%2Fcommutative-ring-theory%2F02819830750568B06C16E6199F3562C1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"> </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> <a href="#CITEREFEisenbud1995">Eisenbud 1995</a>, Exercise A 3.17 </li> <li id="cite_note-FOOTNOTEMilnor19719-20"><a href="#cite_ref-FOOTNOTEMilnor19719_20-0">^</a> <a href="#CITEREFMilnor1971">Milnor (1971)</a>, p. 9. </li> <li id="cite_note-21"><a href="#cite_ref-21">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170116190119/http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_ideals.html">"ideals"</a>. www.math.uiuc.edu. Archived from <a rel="nofollow" class="external text" href="http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_ideals.html">the original</a> on 2017-01-16. Retrieved 2017-01-14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.math.uiuc.edu&rft.atitle=ideals&rft_id=http%3A%2F%2Fwww.math.uiuc.edu%2FMacaulay2%2Fdoc%2FMacaulay2-1.9.2%2Fshare%2Fdoc%2FMacaulay2%2FMacaulay2Doc%2Fhtml%2F_ideals.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170116185903/http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_sums_cm_spproducts_cm_spand_sppowers_spof_spideals.html">"sums, products, and powers of ideals"</a>. www.math.uiuc.edu. Archived from <a rel="nofollow" class="external text" href="http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_sums_cm_spproducts_cm_spand_sppowers_spof_spideals.html">the original</a> on 2017-01-16. Retrieved 2017-01-14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.math.uiuc.edu&rft.atitle=sums%2C+products%2C+and+powers+of+ideals&rft_id=http%3A%2F%2Fwww.math.uiuc.edu%2FMacaulay2%2Fdoc%2FMacaulay2-1.9.2%2Fshare%2Fdoc%2FMacaulay2%2FMacaulay2Doc%2Fhtml%2F_sums_cm_spproducts_cm_spand_sppowers_spof_spideals.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20170116185829/http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_intersection_spof_spideals.html">"intersection of ideals"</a>. www.math.uiuc.edu. Archived from <a rel="nofollow" class="external text" href="http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_intersection_spof_spideals.html">the original</a> on 2017-01-16. Retrieved 2017-01-14.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.math.uiuc.edu&rft.atitle=intersection+of+ideals&rft_id=http%3A%2F%2Fwww.math.uiuc.edu%2FMacaulay2%2Fdoc%2FMacaulay2-1.9.2%2Fshare%2Fdoc%2FMacaulay2%2FMacaulay2Doc%2Fhtml%2F_intersection_spof_spideals.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"> </li> <li id="cite_note-FOOTNOTEAtiyahMacdonald1969Proposition_3.16-24"><a href="#cite_ref-FOOTNOTEAtiyahMacdonald1969Proposition_3.16_24-0">^</a> <a href="#CITEREFAtiyahMacdonald1969">Atiyah & Macdonald (1969)</a>, Proposition 3.16. </li> </ol></div></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAtiyahMacdonald1969" class="citation book cs1"><a href="/wiki/Michael_Atiyah" title="Michael Atiyah">Atiyah, Michael F.</a>; <a href="/wiki/Ian_G._Macdonald" title="Ian G. Macdonald">Macdonald, Ian G.</a> (1969). <a href="/wiki/Introduction_to_Commutative_Algebra" title="Introduction to Commutative Algebra">Introduction to Commutative Algebra</a>. Perseus Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-201-00361-9" title="Special:BookSources/0-201-00361-9"><bdi>0-201-00361-9</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&rft.pub=Perseus+Books&rft.date=1969&rft.isbn=0-201-00361-9&rft.aulast=Atiyah&rft.aufirst=Michael+F.&rft.au=Macdonald%2C+Ian+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDummitFoote2004" class="citation book cs1">Dummit, David Steven; Foote, Richard Martin (2004). Abstract algebra (Third ed.). Hoboken, NJ: John Wiley & Sons, Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780471433347" title="Special:BookSources/9780471433347"><bdi>9780471433347</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Abstract+algebra&rft.place=Hoboken%2C+NJ&rft.edition=Third&rft.pub=John+Wiley+%26+Sons%2C+Inc.&rft.date=2004&rft.isbn=9780471433347&rft.aulast=Dummit&rft.aufirst=David+Steven&rft.au=Foote%2C+Richard+Martin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEisenbud1995" class="citation cs2"><a href="/wiki/David_Eisenbud" title="David Eisenbud">Eisenbud, David</a> (1995), Commutative Algebra with a View toward Algebraic Geometry, <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 150, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <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%2F978-1-4612-5350-1">10.1007/978-1-4612-5350-1</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-94268-1" title="Special:BookSources/978-0-387-94268-1"><bdi>978-0-387-94268-1</bdi></a>, <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1322960">1322960</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Commutative+Algebra+with+a+View+toward+Algebraic+Geometry&rft.place=Berlin%2C+New+York&rft.series=Graduate+Texts+in+Mathematics&rft.pub=Springer-Verlag&rft.date=1995&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1322960%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-5350-1&rft.isbn=978-0-387-94268-1&rft.aulast=Eisenbud&rft.aufirst=David&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLang2005" class="citation book cs1"><a href="/wiki/Serge_Lang" title="Serge Lang">Lang, Serge</a> (2005). Undergraduate Algebra (Third ed.). <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-387-22025-3" title="Special:BookSources/978-0-387-22025-3"><bdi>978-0-387-22025-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Undergraduate+Algebra&rft.edition=Third&rft.pub=Springer-Verlag&rft.date=2005&rft.isbn=978-0-387-22025-3&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHazewinkelGubareniGubareniKirichenko2004" class="citation book cs1"><a href="/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Hazewinkel, Michiel</a>; Gubareni, Nadiya; Gubareni, Nadezhda Mikhaĭlovna; Kirichenko, Vladimir V. (2004). Algebras, rings and modules. 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=Gubareni%2C+Nadezhda+Mikha%C4%ADlovna&rft.au=Kirichenko%2C+Vladimir+V.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMilnor1971" class="citation book cs1"><a href="/wiki/John_Milnor" title="John Milnor">Milnor, John Willard</a> (1971). Introduction to algebraic K-theory. Annals of Mathematics Studies. Vol. 72. Princeton, NJ: <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780691081014" title="Special:BookSources/9780691081014"><bdi>9780691081014</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0349811">0349811</a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0237.18005">0237.18005</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+algebraic+K-theory&rft.place=Princeton%2C+NJ&rft.series=Annals+of+Mathematics+Studies&rft.pub=Princeton+University+Press&rft.date=1971&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0237.18005%23id-name%3DZbl&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0349811%23id-name%3DMR&rft.isbn=9780691081014&rft.aulast=Milnor&rft.aufirst=John+Willard&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Ideal_(ring_theory)&action=edit&section=14" title="Edit section: External links">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevinson2014" class="citation web cs1">Levinson, Jake (July 14, 2014). <a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/866598">"The Geometric Interpretation for Extension of Ideals?"</a>. <a href="/wiki/Stack_Exchange" title="Stack Exchange">Stack Exchange</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stack+Exchange&rft.atitle=The+Geometric+Interpretation+for+Extension+of+Ideals%3F&rft.date=2014-07-14&rft.aulast=Levinson&rft.aufirst=Jake&rft_id=https%3A%2F%2Fmath.stackexchange.com%2Fq%2F866598&rfr_id=info%3Asid%2Fen.wikipedia.org%3AIdeal+%28ring+theory%29" class="Z3988"></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output 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