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class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">přesunout do postranního panelu</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">skrýt</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(úvod)</div> </a> </li> <li id="toc-Definice" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definice"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Definice</span> </div> </a> <ul id="toc-Definice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Příklady_ideálů" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Příklady_ideálů"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Příklady ideálů</span> </div> </a> <ul id="toc-Příklady_ideálů-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Operace_s_ideály" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Operace_s_ideály"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Operace s ideály</span> </div> </a> <ul id="toc-Operace_s_ideály-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vlastnosti" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Vlastnosti"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Vlastnosti</span> </div> </a> <ul id="toc-Vlastnosti-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Věta" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Věta"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Věta</span> </div> </a> <ul id="toc-Věta-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Odkazy" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Odkazy"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Odkazy</span> </div> </a> <button aria-controls="toc-Odkazy-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Přepnout podsekci Odkazy</span> </button> <ul id="toc-Odkazy-sublist" class="vector-toc-list"> <li id="toc-Literatura" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Literatura"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Literatura</span> </div> </a> <ul id="toc-Literatura-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Související_články" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Související_články"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Související články</span> </div> </a> <ul id="toc-Související_články-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Externí_odkazy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Externí_odkazy"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.3</span> <span>Externí odkazy</span> </div> </a> <ul id="toc-Externí_odkazy-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Obsah" 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="Přepnout obsah" > <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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Přepnout obsah</span> </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"><span class="mw-page-title-main">Ideál (teorie okruhů)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Přejděte k článku v jiném jazyce. Je dostupný v 40 jazycích" > <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" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">40 jazyků</span> </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) – arabština" lang="ar" hreflang="ar" data-title="مثالي (نظرية الحلقات)" data-language-autonym="العربية" data-language-local-name="arabština" class="interlanguage-link-target"><span>العربية</span></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) – bulharština" lang="bg" hreflang="bg" data-title="Идеал (теория на пръстените)" data-language-autonym="Български" data-language-local-name="bulharština" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Ideal_(matem%C3%A0tiques)" title="Ideal (matemàtiques) – katalánština" lang="ca" hreflang="ca" data-title="Ideal (matemàtiques)" data-language-autonym="Català" data-language-local-name="katalánština" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%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) – čuvaština" lang="cv" hreflang="cv" data-title="Идеал (алгебра)" data-language-autonym="Чӑвашла" data-language-local-name="čuvaština" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Ideal_(ringteori)" title="Ideal (ringteori) – dánština" lang="da" hreflang="da" data-title="Ideal (ringteori)" data-language-autonym="Dansk" data-language-local-name="dánština" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Ideal_(Ringtheorie)" title="Ideal (Ringtheorie) – němčina" lang="de" hreflang="de" data-title="Ideal (Ringtheorie)" data-language-autonym="Deutsch" data-language-local-name="němčina" class="interlanguage-link-target"><span>Deutsch</span></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) – řečtina" lang="el" hreflang="el" data-title="Ιδεώδες (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="řečtina" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Ideal_(ring_theory)" title="Ideal (ring theory) – angličtina" lang="en" hreflang="en" data-title="Ideal (ring theory)" data-language-autonym="English" data-language-local-name="angličtina" class="interlanguage-link-target"><span>English</span></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"><span>Esperanto</span></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) – španělština" lang="es" hreflang="es" data-title="Ideal (teoría de anillos)" data-language-autonym="Español" data-language-local-name="španělština" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Ringi_ideaal" title="Ringi ideaal – estonština" lang="et" hreflang="et" data-title="Ringi ideaal" data-language-autonym="Eesti" data-language-local-name="estonština" class="interlanguage-link-target"><span>Eesti</span></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) – perština" lang="fa" hreflang="fa" data-title="ایده‌آل (نظریه حلقه‌ها)" data-language-autonym="فارسی" data-language-local-name="perština" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ideaali_(rengasteoria)" title="Ideaali (rengasteoria) – finština" lang="fi" hreflang="fi" data-title="Ideaali (rengasteoria)" data-language-autonym="Suomi" data-language-local-name="finština" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Id%C3%A9al" title="Idéal – francouzština" lang="fr" hreflang="fr" data-title="Idéal" data-language-autonym="Français" data-language-local-name="francouzština" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Ideal_(teor%C3%ADa_dos_aneis)" title="Ideal (teoría dos aneis) – galicijština" lang="gl" hreflang="gl" data-title="Ideal (teoría dos aneis)" data-language-autonym="Galego" data-language-local-name="galicijština" class="interlanguage-link-target"><span>Galego</span></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) – hebrejština" lang="he" hreflang="he" data-title="אידיאל (אלגברה)" data-language-autonym="עברית" data-language-local-name="hebrejština" class="interlanguage-link-target"><span>עברית</span></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) – maďarština" lang="hu" hreflang="hu" data-title="Ideál (gyűrűelmélet)" data-language-autonym="Magyar" data-language-local-name="maďarština" class="interlanguage-link-target"><span>Magyar</span></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"><span>Interlingua</span></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) – indonéština" lang="id" hreflang="id" data-title="Ideal (teori gelanggang)" data-language-autonym="Bahasa Indonesia" data-language-local-name="indonéština" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Ideale_(matematica)" title="Ideale (matematica) – italština" lang="it" hreflang="it" data-title="Ideale (matematica)" data-language-autonym="Italiano" data-language-local-name="italština" class="interlanguage-link-target"><span>Italiano</span></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) – japonština" lang="ja" hreflang="ja" data-title="イデアル (環論)" data-language-autonym="日本語" data-language-local-name="japonština" class="interlanguage-link-target"><span>日本語</span></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) – kazaština" lang="kk" hreflang="kk" data-title="Идеал (математикада)" data-language-autonym="Қазақша" data-language-local-name="kazaština" class="interlanguage-link-target"><span>Қазақша</span></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 – korejština" lang="ko" hreflang="ko" data-title="아이디얼" data-language-autonym="한국어" data-language-local-name="korejština" class="interlanguage-link-target"><span>한국어</span></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) – lombardština" lang="lmo" hreflang="lmo" data-title="Ideaal (matemàtica)" data-language-autonym="Lombard" data-language-local-name="lombardština" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Ideaal_(ringtheorie)" title="Ideaal (ringtheorie) – nizozemština" lang="nl" hreflang="nl" data-title="Ideaal (ringtheorie)" data-language-autonym="Nederlands" data-language-local-name="nizozemština" class="interlanguage-link-target"><span>Nederlands</span></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 – norština (nynorsk)" lang="nn" hreflang="nn" data-title="Ideal i matematikk" data-language-autonym="Norsk nynorsk" data-language-local-name="norština (nynorsk)" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Ideal_(matematikk)" title="Ideal (matematikk) – norština (bokmål)" lang="nb" hreflang="nb" data-title="Ideal (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="norština (bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></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) – polština" lang="pl" hreflang="pl" data-title="Ideał (teoria pierścieni)" data-language-autonym="Polski" data-language-local-name="polština" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Ideal_(teoria_dos_an%C3%A9is)" title="Ideal (teoria dos anéis) – portugalština" lang="pt" hreflang="pt" data-title="Ideal (teoria dos anéis)" data-language-autonym="Português" data-language-local-name="portugalština" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Ideal_(teoria_inelelor)" title="Ideal (teoria inelelor) – rumunština" lang="ro" hreflang="ro" data-title="Ideal (teoria inelelor)" data-language-autonym="Română" data-language-local-name="rumunština" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%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) – ruština" lang="ru" hreflang="ru" data-title="Идеал (алгебра)" data-language-autonym="Русский" data-language-local-name="ruština" class="interlanguage-link-target"><span>Русский</span></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) – slovenština" lang="sk" hreflang="sk" data-title="Ideál (okruhu)" data-language-autonym="Slovenčina" data-language-local-name="slovenština" class="interlanguage-link-target"><span>Slovenčina</span></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) – slovinština" lang="sl" hreflang="sl" data-title="Ideal (teorija kolobarjev)" data-language-autonym="Slovenščina" data-language-local-name="slovinština" class="interlanguage-link-target"><span>Slovenščina</span></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) – srbština" lang="sr" hreflang="sr" data-title="Идеал (математика)" data-language-autonym="Српски / srpski" data-language-local-name="srbština" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Ideal_(ringteori)" title="Ideal (ringteori) – švédština" lang="sv" hreflang="sv" data-title="Ideal (ringteori)" data-language-autonym="Svenska" data-language-local-name="švédština" class="interlanguage-link-target"><span>Svenska</span></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) – tamilština" lang="ta" hreflang="ta" data-title="வளையத்தில் சீர்மம் (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="tamilština" class="interlanguage-link-target"><span>தமிழ்</span></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) – ukrajinština" lang="uk" hreflang="uk" data-title="Ідеал (алгебра)" data-language-autonym="Українська" data-language-local-name="ukrajinština" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a 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data-event-name="pinnable-header.vector-appearance.unpin">skrýt</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Z Wikipedie, otevřené encyklopedie</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="cs" dir="ltr"><p><i><b>Ideál</b></i> je <a href="/wiki/Matematika" title="Matematika">matematický</a> pojem z oblasti <a href="/wiki/Algebra" title="Algebra">algebry</a> označující podmnožinu nějakého <a href="/wiki/Okruh_(algebra)" title="Okruh (algebra)">okruhu</a> s jistými „dobrými“ vlastnostmi. </p> <ul><li>Tak jako <a href="/wiki/Norm%C3%A1ln%C3%AD_podgrupa" title="Normální podgrupa">normální podgrupy</a> jsou speciálními případy <a href="/wiki/Podgrupa" title="Podgrupa">podgrup</a>, jsou rovněž ideály jisté podmnožiny daného <a href="/wiki/Okruh_(algebra)" title="Okruh (algebra)">okruhu</a>.</li></ul> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definice">Definice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&veaction=edit&section=1" title="Editace sekce: Definice" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&action=edit&section=1" title="Editovat zdrojový kód sekce Definice"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Množina <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \emptyset \neq I\subseteq R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∅</mi> <mo>≠</mo> <mi>I</mi> <mo>⊆</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \emptyset \neq I\subseteq R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a77af68f3fc6e300fbe40853b8148612d269abf9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.295ex; height:2.843ex;" alt="{\displaystyle \emptyset \neq I\subseteq R}"></span>, kde <i>R</i> je <a href="/wiki/Okruh_(algebra)" title="Okruh (algebra)">okruh</a>, se nazývá levý resp. pravý <b>ideál</b>, má-li následující vlastnosti: </p> <ul><li>pro každé <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6cfbf3b87c8aa4be682ee15fcd61a7085888fc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.274ex; height:2.509ex;" alt="{\displaystyle a,b\in I}"></span> je také <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a-b\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>−</mo> <mi>b</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a-b\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961721cd7710cf87571c660d94e6e99b223c2b19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.08ex; height:2.343ex;" alt="{\displaystyle a-b\in I}"></span></li> <li>pro každé <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06736171fb6cabbf6888f6c07cfc4630cd799ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.242ex; height:2.176ex;" alt="{\displaystyle a\in I}"></span> a každé <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\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></span><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}"></span> je také <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\cdot a\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>⋅</mo> <mi>a</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\cdot a\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac5cfb01c73a20b3eb8dfabacb818517eaff530" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.97ex; height:2.176ex;" alt="{\displaystyle r\cdot a\in I}"></span> resp. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot r\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>r</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot r\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234665eaddbb641fafcb5c92fbf3e19bea152b05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.97ex; height:2.176ex;" alt="{\displaystyle a\cdot r\in I}"></span></li></ul> <p>Je-li ideál zároveň levý i pravý, nazývá se oboustranný ideál, nebo prostě jen ideál. </p><p>Nechť (<i>R</i>, +, •) je okruh, <i>M</i> je libovolná podmnožina množiny <i>R</i>. Potom průnik všech ideálů v <i>R</i>, které obsahují množinu <i>M</i>, je ideál v <i>R</i>, který se nazývá <b>ideálem generovaným množinou</b> a značí se [<i>M</i>]. Množina <i>M</i> se nazývá <b>systém generátorů</b> ideálu [<i>M</i>] a její prvky <b>generátory</b> tohoto ideálu. </p><p>Prázdná množina generuje v libovolném okruhu nulový ideál <i>R</i>. </p> <div class="mw-heading mw-heading2"><h2 id="Příklady_ideálů"><span id="P.C5.99.C3.ADklady_ide.C3.A1l.C5.AF"></span>Příklady ideálů</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&veaction=edit&section=2" title="Editace sekce: Příklady ideálů" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&action=edit&section=2" title="Editovat zdrojový kód sekce Příklady ideálů"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>V každém okruhu <i>R</i> jsou <a href="/wiki/Mno%C5%BEina" title="Množina">množiny</a> <i>{0}</i> a <i>R</i> ideály. Tyto ideály se nazývají <b>triviální ideály</b> v <i>R</i>. Ideál, který není triviální se nazývá <b>netriviální</b> nebo také <b>vlastní</b>.</li> <li>Každá podmnožina tvaru <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (a)=\{a\cdot r;r\in R\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>a</mi> <mo>⋅</mo> <mi>r</mi> <mo>;</mo> <mi>r</mi> <mo>∈</mo> <mi>R</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (a)=\{a\cdot r;r\in R\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0b5e847e98fcbc15c2034967e1dacd35c2e3952" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.107ex; height:2.843ex;" alt="{\displaystyle (a)=\{a\cdot r;r\in R\}}"></span> je ideál v <i>R</i>. Ideály tvaru <i>(a)</i> se nazývají <b><a href="/wiki/Hlavn%C3%AD_ide%C3%A1l_(teorie_okruh%C5%AF)" title="Hlavní ideál (teorie okruhů)">hlavní ideály</a></b> v <i>R</i>.</li> <li>V okruhu <a href="/wiki/Cel%C3%A9_%C4%8D%C3%ADslo" title="Celé číslo">celých čísel</a> je množina všech <a href="/wiki/Sud%C3%A1_a_lich%C3%A1_%C4%8D%C3%ADsla" title="Sudá a lichá čísla">sudých čísel</a> ideálem, konkrétně hlavním ideálem (2).</li> <li>Libovolný podokruh komutativního okruhu nemusí být jeho ideálem. Například v okruhu racionálních čísel (<i>Q</i>,+,•) tvoří celá čísla podokruh (<i>Z</i>,+,•). Ten však není ideálem v <b>Q</b>, neboť nesplňuje podmínku: pro každé <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06736171fb6cabbf6888f6c07cfc4630cd799ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.242ex; height:2.176ex;" alt="{\displaystyle a\in I}"></span> a každé <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\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></span><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}"></span> je také <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\cdot a\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>⋅</mo> <mi>a</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\cdot a\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac5cfb01c73a20b3eb8dfabacb818517eaff530" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.97ex; height:2.176ex;" alt="{\displaystyle r\cdot a\in I}"></span> resp. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot r\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>r</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot r\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234665eaddbb641fafcb5c92fbf3e19bea152b05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.97ex; height:2.176ex;" alt="{\displaystyle a\cdot r\in I}"></span>. Stačí volit třeba <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=3,r={\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mi>r</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=3,r={\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/808fe07c512c350d5a26143fe63281be264a0bec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.67ex; height:5.176ex;" alt="{\displaystyle a=3,r={\frac {1}{2}}}"></span>, pak <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\in Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>∈</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\in Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86036401e8ff797815f2c42e8aaca07fb43fba13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.684ex; height:2.176ex;" alt="{\displaystyle 3\in Z}"></span> a <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3\cdot {\frac {1}{2}}={\frac {3}{2}}\notin Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>∉</mo> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3\cdot {\frac {1}{2}}={\frac {3}{2}}\notin Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03d80960f244a2283323588041b74f33f264d1f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:14.458ex; height:5.176ex;" alt="{\displaystyle 3\cdot {\frac {1}{2}}={\frac {3}{2}}\notin Z}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Operace_s_ideály"><span id="Operace_s_ide.C3.A1ly"></span>Operace s ideály</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&veaction=edit&section=3" title="Editace sekce: Operace s ideály" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&action=edit&section=3" title="Editovat zdrojový kód sekce Operace s ideály"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><b><a href="/wiki/Pr%C5%AFnik" title="Průnik">průnik</a></b> ideálů <i>I,J</i> je ideál <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\cap 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\cap J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9444054edd640576f42e6bd9700079279bcf022" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.226ex; height:2.176ex;" alt="{\displaystyle I\cap J}"></span>, který je největším ideálem, obsaženým v obou ideálech <i>I,J</i>.</li> <li><b>součet</b> ideálů <i>I,J</i> je ideál <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,I+J=\{i+j;i\in I,j\in J\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mi>I</mi> <mo>+</mo> <mi>J</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>i</mi> <mo>+</mo> <mi>j</mi> <mo>;</mo> <mi>i</mi> <mo>∈</mo> <mi>I</mi> <mo>,</mo> <mi>j</mi> <mo>∈</mo> <mi>J</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,I+J=\{i+j;i\in I,j\in J\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2bf26d5af6182d08fc3df7acd55042132772968" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.048ex; height:2.843ex;" alt="{\displaystyle \,I+J=\{i+j;i\in I,j\in J\}}"></span>, který je nejmenším ideálem obsahujícím oba ideály <i>I,J</i>.</li> <li><b>součin</b> ideálů <i>I,J</i> je ideál <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\cdot J=\{\sum _{k=1}^{n}a_{k}\cdot b_{k};n\in N,a_{k}\in I,b_{k}\in J\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>⋅</mo> <mi>J</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⋅</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>;</mo> <mi>n</mi> <mo>∈</mo> <mi>N</mi> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mi>I</mi> <mo>,</mo> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mi>J</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I\cdot J=\{\sum _{k=1}^{n}a_{k}\cdot b_{k};n\in N,a_{k}\in I,b_{k}\in J\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afd54a5e36e87964ea4ecfda097826fe102e3e09" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:41.702ex; height:6.843ex;" alt="{\displaystyle I\cdot J=\{\sum _{k=1}^{n}a_{k}\cdot b_{k};n\in N,a_{k}\in I,b_{k}\in J\}}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Vlastnosti">Vlastnosti</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&veaction=edit&section=4" title="Editace sekce: Vlastnosti" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&action=edit&section=4" title="Editovat zdrojový kód sekce Vlastnosti"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Ideál <i>I</i> v okruhu <i>R</i> se nazývá <b>maximální ideál</b>, je-li <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\neq 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\neq R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe38b8d0d6ca989337107df023f9782afae6e014" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.034ex; height:2.676ex;" alt="{\displaystyle I\neq R}"></span> a pro každý ideál <i>J</i>, že <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I\subseteq 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\subseteq J}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16b159c54aac97dc2112a26c8ad64655f7697fb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.742ex; height:2.343ex;" alt="{\displaystyle I\subseteq J}"></span>, je <i>I = J</i> nebo <i>J = R</i>.</li> <li>Ideál <i>I</i> v okruhu <i>R</i> se nazývá <b><a href="/wiki/Prvoide%C3%A1l_(teorie_okruh%C5%AF)" title="Prvoideál (teorie okruhů)">prvoideál</a></b>, jestliže pro každé <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a,b\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d992800adce8f580e87a3c79b62f0e12d5349e1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.866ex; height:2.509ex;" alt="{\displaystyle a,b\in R}"></span> takové, že <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\cdot b\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>⋅</mo> <mi>b</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\cdot b\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3af62231520b97c6d875f2198c70744e10e0b3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.919ex; height:2.176ex;" alt="{\displaystyle a\cdot b\in I}"></span>, je buďto <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c06736171fb6cabbf6888f6c07cfc4630cd799ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.242ex; height:2.176ex;" alt="{\displaystyle a\in I}"></span> nebo <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4535d2ecfbd140eb8908c10eba1d79f17bd7c0ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.01ex; height:2.176ex;" alt="{\displaystyle b\in I}"></span>.</li> <li>Jsou-li <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1},a_{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1},a_{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3483c81917a179212dbac7cd49e52581085da3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.249ex; height:2.009ex;" alt="{\displaystyle a_{1},a_{2},}"></span> … , <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05e256a120c3ab9f8958de71acdf81cd75065e3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.319ex; height:2.009ex;" alt="{\displaystyle a_{k}}"></span> libovolné prvky z ideálu <i>I</i> v okruhu <i>R</i>, je každá jejich lineární kombinace s koeficienty z <i>R</i> prvkem ideálu <i>I</i>, tj. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\forall r_{1},r_{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\forall r_{1},r_{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/110393a17fbc4d626ff0b7bb7a4eb5ca92e53423" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.084ex; height:2.843ex;" alt="{\displaystyle (\forall r_{1},r_{2},}"></span>… , <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{k}\in R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{k}\in R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/673c15f80a77523139e597dae00411bbdcea9df4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.647ex; height:2.843ex;" alt="{\displaystyle r_{k}\in R)}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}r_{1}+a_{2}r_{2}+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}r_{1}+a_{2}r_{2}+}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f791244000191c9fd0b6e1df9367379743e035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.422ex; height:2.343ex;" alt="{\displaystyle a_{1}r_{1}+a_{2}r_{2}+}"></span>… <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +a_{k}r_{k}\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +a_{k}r_{k}\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb3e5d9b7b62c77674b2c61809b53ef43c59265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.276ex; height:2.509ex;" alt="{\displaystyle +a_{k}r_{k}\in I}"></span>.</li></ul> <p><b>Příklad:</b> </p><p>V okruhu celých čísel <b>Z</b> máme určit ideál <i>I</i> = [96, 14]. Snažíme se v tomto ideálu najít nenulové číslo s co nejmenší <a href="/wiki/Absolutn%C3%AD_hodnota" title="Absolutní hodnota">absolutní hodnotou</a>. Musí být 1 • 96 + (- 6) •14 = 12 ∈ <i>I</i> a též 1 • 14 + (- 1) •12 = 2 ∈ <i>I</i> . Podle druhé podmínky (viz výše) obsahuje <i>I</i> všechny celočíselné násobky čísla 2, tj. všechna sudá čísla. Protože podle definice ideálu (Podmnožina <i>I</i> okruhu <b>R</b> je ideálem v právě tehdy, když je neprázdná a platí pro ni podmínky viz výše) množina všech sudých čísel tvoří zřejmě ideál v <b>Z</b>, je <i>I</i> = {..., -6, -4, -2, 0, 2, 4, 6, ...}. </p><p>Týž ideál může mít různé systémy generátorů. Např. ideál <i>I</i> z předchozího příkladu je generován číslem 2, tj. <i>I</i> = [2], a též například <i>I</i> = [6, 8, -10]. </p><p>Platí věta: <i>Každý maximální ideál je prvoideál.</i> Opačné tvrzení v obecném případě neplatí, tj. existují prvoideály, které nejsou maximální. Pokud však <i>R</i> je číselný okruh (tj. podokruh okruhu <a href="/wiki/Komplexn%C3%AD_%C4%8D%C3%ADslo" title="Komplexní číslo">komplexních</a> algebraických celých čísel), je každý prvoideál v <i>R</i> maximálním ideálem. </p> <ul><li>Ideály jsou právě ty množiny, <a href="/wiki/Faktorizace" title="Faktorizace">faktorizací</a> podle nichž vznikne z okruhu opět <a href="/wiki/Okruh_(algebra)" title="Okruh (algebra)">okruh</a>.</li> <li>Prvoideály jsou právě ty množiny, faktorizací podle nichž vznikne z okruhu <a href="/wiki/Obor_integrity" title="Obor integrity">obor integrity</a>.</li> <li>Maximální ideály jsou právě ty množiny, faktorizací podle nichž vznikne <a href="/wiki/T%C4%9Bleso_(algebra)" title="Těleso (algebra)">těleso</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Věta"><span id="V.C4.9Bta"></span>Věta</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&veaction=edit&section=5" title="Editace sekce: Věta" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&action=edit&section=5" title="Editovat zdrojový kód sekce Věta"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Nechť <i>R</i> je okruh s jednotkovým prvkem a nechť <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=\{a_{1},a_{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=\{a_{1},a_{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15083e099e7a54ef4886d5500db73260af122c6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.952ex; height:2.843ex;" alt="{\displaystyle M=\{a_{1},a_{2},}"></span>… <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ,a_{k}\subseteq R\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>⊆</mo> <mi>R</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ,a_{k}\subseteq R\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2c25ce5c3ecff43adf6172ac2b1b9fb6bca7067" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.377ex; height:2.843ex;" alt="{\displaystyle ,a_{k}\subseteq R\}}"></span>. Pak ideál [<i>M</i>] se skládá právě ze všech prvků tvaru <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\forall r_{1},r_{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi mathvariant="normal">∀</mi> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\forall r_{1},r_{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/110393a17fbc4d626ff0b7bb7a4eb5ca92e53423" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.084ex; height:2.843ex;" alt="{\displaystyle (\forall r_{1},r_{2},}"></span>… , <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{k}\in R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r_{k}\in R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/673c15f80a77523139e597dae00411bbdcea9df4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.647ex; height:2.843ex;" alt="{\displaystyle r_{k}\in R)}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1}r_{1}+a_{2}r_{2}+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{1}r_{1}+a_{2}r_{2}+}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52f791244000191c9fd0b6e1df9367379743e035" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.422ex; height:2.343ex;" alt="{\displaystyle a_{1}r_{1}+a_{2}r_{2}+}"></span>… <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +a_{k}r_{k}\in I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +a_{k}r_{k}\in I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bb3e5d9b7b62c77674b2c61809b53ef43c59265" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.276ex; height:2.509ex;" alt="{\displaystyle +a_{k}r_{k}\in I}"></span>, tj. [<i>M</i>] = <i>I</i>, kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=\{a_{1}r_{1}+a_{2}r_{2}+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\{a_{1}r_{1}+a_{2}r_{2}+}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb2fdcd4437271069501c327cf8196e2d1b801f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.855ex; height:2.843ex;" alt="{\displaystyle I=\{a_{1}r_{1}+a_{2}r_{2}+}"></span>… <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +a_{k}r_{k};r_{1},r_{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +a_{k}r_{k};r_{1},r_{2},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df2f06150606a1803cf1ec8bfab1a9c3a19abf46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.185ex; height:2.343ex;" alt="{\displaystyle +a_{k}r_{k};r_{1},r_{2},}"></span>… <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ,r_{k}\in R\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>,</mo> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>∈</mo> <mi>R</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ,r_{k}\in R\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8801f2bc198bebfa336154858208ae822f24b85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.938ex; height:2.843ex;" alt="{\displaystyle ,r_{k}\in R\}}"></span>. </p><p><b>Příklad užití této věty</b> </p><p>V okruhu <b>Z</b>[<i>x</i>] polynomů jedné neurčité s celočíselnými koeficienty máme sestrojit ideál [<i>x</i>, 2]. Podle věty výše (v <b>Z</b>[<i>x</i>] existuje jednotkový prvek) se tento ideál skládá ze všech prvků tvaru: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\cdot f_{1}(x)+2\cdot f_{2}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>⋅</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x\cdot f_{1}(x)+2\cdot f_{2}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1af27d7f3180fb7a0f53043f446f378a89a84e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.356ex; height:2.843ex;" alt="{\displaystyle x\cdot f_{1}(x)+2\cdot f_{2}(x)}"></span> kde <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}(x),f_{2}(x)\in Z[x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>Z</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}(x),f_{2}(x)\in Z[x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d815586390247737c6bdde5ee5c51a17aebf5626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.843ex; height:2.843ex;" alt="{\displaystyle f_{1}(x),f_{2}(x)\in Z[x]}"></span>. </p><p>Tedy [<i>x</i>, 2] je množina všech polynomů <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0}+a_{1}x+}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>x</mi> <mo>+</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{0}+a_{1}x+}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b925abf7be5f263cf01906e8aab2158160c5038" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.546ex; height:2.343ex;" alt="{\displaystyle a_{0}+a_{1}x+}"></span>… <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +a_{x}x^{n}\in Z[x]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>∈</mo> <mi>Z</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +a_{x}x^{n}\in Z[x]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e3c5a2f333514a0c6ae5d7ef67fbcacb4e6213f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.903ex; height:2.843ex;" alt="{\displaystyle +a_{x}x^{n}\in Z[x]}"></span>, jejíž člen <i>a</i><sub>0</sub> je sudé číslo. Ideál [<i>x</i>, 2] je tudíž vlastní podmnožina v <b>Z</b>[<i>x</i>]. </p> <div class="mw-heading mw-heading2"><h2 id="Odkazy">Odkazy</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&veaction=edit&section=6" title="Editace sekce: Odkazy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&action=edit&section=6" title="Editovat zdrojový kód sekce Odkazy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Literatura">Literatura</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&veaction=edit&section=7" title="Editace sekce: Literatura" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&action=edit&section=7" title="Editovat zdrojový kód sekce Literatura"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>BLAŽEK, Jaroslav, KOMAN, Milan a VOJTÁŠKOVÁ, Blanka. <i>Algebra a teoretická aritmetika</i>. 1. vyd. Praha: Státní pedagogické nakladatelství, 1985, 258 s. Učebnice pro vysoké školy (Státní pedagogické nakladatelství).</li></ul> <div class="mw-heading mw-heading3"><h3 id="Související_články"><span id="Souvisej.C3.ADc.C3.AD_.C4.8Dl.C3.A1nky"></span>Související články</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&veaction=edit&section=8" title="Editace sekce: Související články" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&action=edit&section=8" title="Editovat zdrojový kód sekce Související články"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Ide%C3%A1l_(teorie_uspo%C5%99%C3%A1d%C3%A1n%C3%AD)" title="Ideál (teorie uspořádání)">Ideál (teorie uspořádání)</a></li> <li><a href="/wiki/Ireducibiln%C3%AD_ide%C3%A1l" title="Ireducibilní ideál">Ireducibilní ideál</a></li> <li><a href="/wiki/Norm%C3%A1ln%C3%AD_podgrupa" title="Normální podgrupa">Normální podgrupa</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Externí_odkazy"><span id="Extern.C3.AD_odkazy"></span>Externí odkazy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&veaction=edit&section=9" title="Editace sekce: Externí odkazy" class="mw-editsection-visualeditor"><span>editovat</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Ide%C3%A1l_(teorie_okruh%C5%AF)&action=edit&section=9" title="Editovat zdrojový kód sekce Externí odkazy"><span>editovat zdroj</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="wd"><span class="sisterproject sisterproject-commons"><span class="sisterproject_image"><span typeof="mw:File"><a href="/wiki/Wikimedia_Commons" title="Wikimedia Commons"><img alt="Logo Wikimedia Commons" 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