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<title>Isomorphiesatz</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Isomorphiesatz</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Die <b>Isomorphiesätze</b> sind zwei <a href="Mathematik" title="Mathematik">mathematische</a> Sätze, die Aussagen über <a href="Gruppe_(Mathematik)" title="Gruppe (Mathematik)">Gruppen</a> machen. Sie lassen sich auch auf komplexere <a href="Algebraische_Struktur" title="Algebraische Struktur">algebraische Strukturen</a> übertragen und sind somit ein wichtiges Resultat der <a href="Universelle_Algebra" title="Universelle Algebra">universellen Algebra</a>. Die Isomorphiesätze sind eine direkte Folgerung aus dem <a href="Homomorphiesatz" title="Homomorphiesatz">Homomorphiesatz</a> der entsprechenden <a href="Algebraische_Struktur" title="Algebraische Struktur">algebraischen Struktur</a>.

</p><p>Manchmal wird der <a href="Homomorphiesatz" title="Homomorphiesatz">Homomorphiesatz</a> als erster Isomorphiesatz bezeichnet. Die unten angegebenen Sätze heißen dann dementsprechend <i>zweiter</i> bzw. <i>dritter Isomorphiesatz</i>.

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<div class="mw-heading mw-heading2"><h2 id="Geschichte">Geschichte</h2></div>

<p>Die Isomorphiesätze wurden zunächst in allgemeiner Form für <a href="Modulhomomorphismus" title="Modulhomomorphismus">Modulhomomorphismen</a> von <a href="Emmy_Noether" title="Emmy Noether">Emmy Noether</a> in ihrer Arbeit <i>Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern</i>, welche 1927 in <a href="Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a> veröffentlicht wurde.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Weniger allgemeine Resultate dieser Sätze lassen sich auch in vorangegangenen Arbeiten von <a href="Richard_Dedekind" title="Richard Dedekind">Richard Dedekind</a> und Emmy Noether finden.

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<div class="mw-heading mw-heading2"><h2 id="Gruppentheorie">Gruppentheorie</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Erster_Isomorphiesatz">Erster Isomorphiesatz</h3></div>

<p>Es seien <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> eine Gruppe, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> ein <a href="Normalteiler" title="Normalteiler">Normalteiler</a> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

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<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> eine <a href="Untergruppe" title="Untergruppe">Untergruppe</a> von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span>. Dann ist auch das <a href="Komplexprodukt" title="Komplexprodukt">Komplexprodukt</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle HN:=\{hn\mid h\in H,n\in N\}}">

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<annotation encoding="application/x-tex">{\displaystyle HN:=\{hn\mid h\in H,n\in N\}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3d2332c1672fb76f468ab6e9aee10cfa344b7a74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.444ex; height:2.843ex;" alt="{\displaystyle HN:=\{hn\mid h\in H,n\in N\}}" loading="lazy"></span> eine Untergruppe von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> ist ein Normalteiler in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle HN}">

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<annotation encoding="application/x-tex">{\displaystyle HN}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ccfec31dfa1055781e20094b0350cd4bfc7c6eca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.127ex; height:2.176ex;" alt="{\displaystyle HN}" loading="lazy"></span> und die Gruppe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\cap N}">

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<annotation encoding="application/x-tex">{\displaystyle H\cap N}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/37535e9f8df7ca6c501ae9373202f19755b9ce2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.71ex; height:2.176ex;" alt="{\displaystyle H\cap N}" loading="lazy"></span> ist ein Normalteiler in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

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<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span>. Es gilt:

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<annotation encoding="application/x-tex">{\displaystyle H/(H\cap N)\cong HN/N.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1af24d7620ce447414d1297113e971871c6328e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.844ex; height:2.843ex;" alt="{\displaystyle H/(H\cap N)\cong HN/N.}" loading="lazy"></span></dd></dl>

<p>Dabei bezeichnet <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cong }">

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<annotation encoding="application/x-tex">{\displaystyle \cong }</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a725ebc5ab8de11d7b71a8aa5a3706c2ea467885.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.049ex; margin-bottom: -0.22ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \cong }" loading="lazy"></span> die <a href="Isomorphismus" title="Isomorphismus">Isomorphie</a> von Gruppen.

</p><p>Der Isomorphismus, der dabei üblicherweise gemeint ist, wird als <i>kanonischer</i> Isomorphismus bezeichnet. Er wird gemäß dem <a href="Homomorphiesatz" title="Homomorphiesatz">Homomorphiesatz</a> von der surjektiven Abbildung

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<annotation encoding="application/x-tex">{\displaystyle f\colon H\to HN/N,\quad h\mapsto hN,}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d579059f91a55d0d00e151dd685dd8b90ba5928.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.703ex; height:2.843ex;" alt="{\displaystyle f\colon H\to HN/N,\quad h\mapsto hN,}" loading="lazy"></span></dd></dl>

<p>induziert, denn es gilt offenbar

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {kern} \left(f\right)=\left\{a\in H\mid aN=N\right\}=\left\{a\in H\mid a\in N\right\}=H\cap N}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/233c28feb80d6c0212123b2c66cc47624070ca53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:58.939ex; height:2.843ex;" alt="{\displaystyle \operatorname {kern} \left(f\right)=\left\{a\in H\mid aN=N\right\}=\left\{a\in H\mid a\in N\right\}=H\cap N}" loading="lazy"></span>.</dd></dl>

<p>Aus dem ersten Isomorphiesatz erhält man als Spezialfall die anschauliche Aussage, dass man genau dann mit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> „erweitern“ darf, wenn <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\cap N=\{0\}}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle H\cap N=\{0\}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4bd400c5e27387005de0438db96201482d9001d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.296ex; height:2.843ex;" alt="{\displaystyle H\cap N=\{0\}}" loading="lazy"></span>.

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<div class="mw-heading mw-heading3"><h3 id="Zweiter_Isomorphiesatz">Zweiter Isomorphiesatz</h3></div>

<p>Es seien <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> eine Gruppe, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span> ein Normalteiler in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> eine Untergruppe von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

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<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy"></span>, die Normalteiler in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"></span> ist. Dann gilt:

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<annotation encoding="application/x-tex">{\displaystyle (G/N)/(H/N)\cong G/H.}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34dc0e6853434d6b428e436958b36300b89ed93b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.922ex; height:2.843ex;" alt="{\displaystyle (G/N)/(H/N)\cong G/H.}" loading="lazy"></span></dd></dl>

<p>In diesem Fall kann man kanonische Isomorphismen in beide Richtungen angeben, einerseits induziert durch

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<annotation encoding="application/x-tex">{\displaystyle G/N\to G/H,\quad gN\mapsto gH,}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e435f2ca06b54c573b8817c82f922a000da63dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.696ex; height:2.843ex;" alt="{\displaystyle G/N\to G/H,\quad gN\mapsto gH,}" loading="lazy"></span></dd></dl>

<p>andererseits durch

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<annotation encoding="application/x-tex">{\displaystyle G\to (G/N)/(H/N),\quad g\mapsto gN(H/N).}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cda7d7d61191f5d60225b37fe813d57aa5d738cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.576ex; height:2.843ex;" alt="{\displaystyle G\to (G/N)/(H/N),\quad g\mapsto gN(H/N).}" loading="lazy"></span></dd></dl>

<p>Anschaulich ausgedrückt besagt der zweite Isomorphiesatz, dass man <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> „kürzen“ darf.

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<div class="mw-heading mw-heading2"><h2 id="Ringe">Ringe</h2></div>

<p>In angepasster Form gelten die Isomorphiesätze auch für Ringe:

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<div class="mw-heading mw-heading3"><h3 id="Erster_Isomorphiesatz_2">Erster Isomorphiesatz</h3></div>

<p>Es seien <span class="mwe-math-element mwe-math-element-inline"><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}">

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<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" 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}" loading="lazy"></span> ein Ring, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/16f656feeddb5d98500bb4d3fc31038d0b87484b.svg" 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}}}" loading="lazy"></span> ein Ideal von <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" 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}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" 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}" loading="lazy"></span> ein Unterring von <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" 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}" loading="lazy"></span>. Dann ist die Summe <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S+{\mathfrak {a}}=\{s+a\mid s\in S,a\in {\mathfrak {a}}\}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S+{\mathfrak {a}}=\{s+a\mid s\in S,a\in {\mathfrak {a}}\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/29bcf4238934518738b1c2648b9bf63e95d8b352.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.72ex; height:2.843ex;" alt="{\displaystyle S+{\mathfrak {a}}=\{s+a\mid s\in S,a\in {\mathfrak {a}}\}}" loading="lazy"></span> ein Unterring von <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" 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}" loading="lazy"></span> und der Schnitt <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\cap {\mathfrak {a}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\cap {\mathfrak {a}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0dd16fd8b46a7432367c36906c756518b885ed78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.244ex; height:2.176ex;" alt="{\displaystyle S\cap {\mathfrak {a}}}" loading="lazy"></span> ein Ideal von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.svg" 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}" loading="lazy"></span>. Es gilt:

</p>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S/(S\cap {\mathfrak {a}})\cong (S+{\mathfrak {a}})/{\mathfrak {a}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b11eeb941289956ae8de15a7a1268fe3a74f8cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.45ex; height:2.843ex;" alt="{\displaystyle S/(S\cap {\mathfrak {a}})\cong (S+{\mathfrak {a}})/{\mathfrak {a}}}" loading="lazy"></span>.</dd></dl>

<p>Dabei bezeichnet <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cong }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cong }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a725ebc5ab8de11d7b71a8aa5a3706c2ea467885.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.049ex; margin-bottom: -0.22ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \cong }" loading="lazy"></span> die Isomorphie von Ringen.

</p>

<div class="mw-heading mw-heading3"><h3 id="Zweiter_Isomorphiesatz_2">Zweiter Isomorphiesatz</h3></div>

<p>Es seien <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" 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}" loading="lazy"></span> ein Ring, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}\subseteq {\mathfrak {a}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {b}}\subseteq {\mathfrak {a}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ce9468640dc9af2ef28a8284dd2718d9475676c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.454ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {b}}\subseteq {\mathfrak {a}}}" loading="lazy"></span> zwei Ideale von <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" 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}" loading="lazy"></span>. Dann ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}/{\mathfrak {b}}=\{a+{\mathfrak {b}}\mid a\in {\mathfrak {a}}\}}">

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}/{\mathfrak {b}}=\{a+{\mathfrak {b}}\mid a\in {\mathfrak {a}}\}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/df50bd79a884d0978ee2d3db89fb5b3e02feed51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.374ex; height:2.843ex;" alt="{\displaystyle {\mathfrak {a}}/{\mathfrak {b}}=\{a+{\mathfrak {b}}\mid a\in {\mathfrak {a}}\}}" loading="lazy"></span> ein Ideal von <span class="mwe-math-element mwe-math-element-inline"><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/{\mathfrak {b}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R/{\mathfrak {b}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02d97fbde298e8160130af983ea6adf40c50aef4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.119ex; height:2.843ex;" alt="{\displaystyle R/{\mathfrak {b}}}" loading="lazy"></span>. Es gilt:

</p>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (R/{\mathfrak {b}})/({\mathfrak {a}}/{\mathfrak {b}})\cong R/{\mathfrak {a}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a283fb133fbe20d6dc7173ee6c19ad7db25ce6f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.605ex; height:2.843ex;" alt="{\displaystyle (R/{\mathfrak {b}})/({\mathfrak {a}}/{\mathfrak {b}})\cong R/{\mathfrak {a}}}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Vektorräume,_abelsche_Gruppen_oder_Objekte_einer_beliebigen_abelschen_Kategorie"><span id="Vektorr.C3.A4ume.2C_abelsche_Gruppen_oder_Objekte_einer_beliebigen_abelschen_Kategorie"></span>Vektorräume, abelsche Gruppen oder Objekte einer beliebigen abelschen Kategorie</h2></div>

<p>Es seien <span class="mwe-math-element mwe-math-element-inline"><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,N\subseteq Q\subseteq P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M,N\subseteq Q\subseteq P}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3dfcbfe199940d49cb015f53ed81206a72417aa9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.32ex; height:2.509ex;" alt="{\displaystyle M,N\subseteq Q\subseteq P}" loading="lazy"></span>

</p>

<ul><li><a href="Vektorraum" title="Vektorraum">Vektorräume</a> über einem <a href="K%C3%B6rpertheorie" class="mw-redirect" title="Körpertheorie">Körper</a></li>

<li>oder <a href="Abelsche_Gruppe" title="Abelsche Gruppe">abelsche Gruppen</a></li>

<li>oder allgemeiner <a href="Modul_(Mathematik)" title="Modul (Mathematik)">Moduln</a> über einem <a href="Ring_(Algebra)" title="Ring (Algebra)">Ring</a></li>

<li>oder ganz allgemein Objekte einer <a href="Abelsche_Kategorie" title="Abelsche Kategorie">abelschen Kategorie</a>.</li></ul>

<p>Dann gilt:

</p>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M/(M\cap N)\cong (M+N)/N}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4332fe367dbfd210976971c7bfacd9b26c3cd457.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.982ex; height:2.843ex;" alt="{\displaystyle M/(M\cap N)\cong (M+N)/N}" loading="lazy"></span></li>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (P/N)/(Q/N)\cong P/Q}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ee351426a502b6cb5b330ab265390af87e85a235.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.662ex; height:2.843ex;" alt="{\displaystyle (P/N)/(Q/N)\cong P/Q}" loading="lazy"></span></li></ul>

<p>Auch hier steht das Symbol <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cong }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \cong }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a725ebc5ab8de11d7b71a8aa5a3706c2ea467885.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.049ex; margin-bottom: -0.22ex; width:1.808ex; height:1.843ex;" alt="{\displaystyle \cong }" loading="lazy"></span> für die Isomorphie der entsprechenden <a href="Algebraische_Struktur" title="Algebraische Struktur">algebraischen Strukturen</a> bzw. Objekte in der jeweiligen Kategorie.

</p><p>Die kanonischen Isomorphismen sind eindeutig dadurch bestimmt, dass sie mit den beiden kanonischen Pfeilen von <span class="mwe-math-element mwe-math-element-inline"><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}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" 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}" loading="lazy"></span> bzw. <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> kompatibel sind.

</p><p>Eine weitreichende Verallgemeinerung der Isomorphiesätze liefert das <a href="Schlangenlemma" title="Schlangenlemma">Schlangenlemma</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Siegfried_Bosch" title="Siegfried Bosch">Siegfried Bosch</a>: <i>Algebra.</i> 8. Auflage. Springer, Berlin/Heidelberg 2013, ISBN 978-3-642-39566-6, Kapitel 1.2.</li>

<li>Christian Karpfinger, Kurt Meyberg: <i>Algebra.</i> 3. Auflage. Springer, Berlin/Heidelberg 2013, ISBN 978-3-8274-3011-3, Kapitel 4.6.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a href="Matroids_Matheplanet" title="Matroids Matheplanet">matheplanet.com</a>: <i><a rel="nofollow" class="external text" href="https://matheplanet.com/matheplanet/nuke/html/article.php?sid=668">Gruppenzwang IV.</a></i> – Ausführliche Erklärungen und Beweise der Isomorphiesätze</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Emmy Noether: <cite style="font-style:italic">Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern</cite>. In: <cite style="font-style:italic">Mathematische Annalen</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>96</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>1</span>, Dezember 1927, <a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220025-5831%22&key=cql">0025-5831</a></span>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>26–61</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/BF01209152">10.1007/BF01209152</a></span> (<a rel="nofollow" class="external text" href="http://link.springer.com/10.1007/BF01209152">springer.com</a> [abgerufen am 13. Mai 2024]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Isomorphiesatz&rft.atitle=Abstrakter+Aufbau+der+Idealtheorie+in+algebraischen+Zahl-+und+Funktionenk%C3%B6rpern&rft.au=Emmy+Noether&rft.date=1927-12&rft.doi=10.1007%2FBF01209152&rft.genre=journal&rft.issn=0025-5831&rft.issue=1&rft.jtitle=Mathematische+Annalen&rft.pages=26-61&rft.volume=96" style="display:none"> </span></span>

</li>

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