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<title>Modulhomomorphismus</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Modulhomomorphismus</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">In der Mathematik ist ein Modulhomomorphismus eine Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\rightarrow N}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/190f9ff6f9f4acba07624a16eb6be3fc9b5a96bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\rightarrow N}" loading="lazy"> zwischen zwei <a href="Modul_(Mathematik)" title="Modul (Mathematik)">Moduln</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><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"> und <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><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"> über einem <a href="Ring_(Algebra)" title="Ring (Algebra)">Ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</math><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">, welche mit der Modulstruktur verträglich ist.

Sie übersetzt beispielsweise die Addition von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><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"> in die Addition von <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><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">.

Eine Addition kann man zweifach übersetzen.

<ol><li>Man addiert zunächst in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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

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</math><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"> und übersetzt dann mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">.</li>

<li>Man übersetzt mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> die Summanden und berechnet die Summe in <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><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">.</li></ol>

Bei einem Homomorphismus ergibt sich stets dasselbe. Ersetzt man in der Definition der <a href="Lineare_Abbildung" title="Lineare Abbildung">linearen Abbildung</a> zwischen Vektorräumen den <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> durch einen Ring, erhält man einen Modulhomomorphismus. Der Ring braucht nicht kommutativ zu sein.

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

<div class="mw-heading mw-heading3"><h3 id="Definition">Definition</h3></div>

Es seien zwei Rechtsmoduln <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,N}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30012178e674e69a55fee64e9c221dbdb0d95b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle M,N}" loading="lazy"> über einem Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</math><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"> gegeben. Eine Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\rightarrow N}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/190f9ff6f9f4acba07624a16eb6be3fc9b5a96bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\rightarrow N}" loading="lazy"> heißt Homomorphismus von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

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</mrow>

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

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</math><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"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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</mrow>

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

</semantics>

</math><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">, wenn für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1},m_{2}\in M}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{1},m_{2}\in M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05c2d2ccf329e110868d1a76cce2f4fc3478a0ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.506ex; height:2.509ex;" alt="{\displaystyle m_{1},m_{2}\in M}" loading="lazy"> und alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in R}">

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</mrow>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca49c66b5e9b5f32249a737e4429c3df136c33f0.svg" 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}" loading="lazy"> gilt:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(m_{1}+m_{2})=f(m_{1})+f(m_{2})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70f2ecb34ba94f947aeaa8cd7617a3ffb9341c89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.421ex; height:2.843ex;" alt="{\displaystyle f(m_{1}+m_{2})=f(m_{1})+f(m_{2})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(m_{1}\cdot r)=f(m_{1})\cdot r}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(m_{1}\cdot r)=f(m_{1})\cdot r}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/95f5dc63509613c0523421d2b7133fe4cdfc0d76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.919ex; height:2.843ex;" alt="{\displaystyle f(m_{1}\cdot r)=f(m_{1})\cdot r}" loading="lazy"></dd></dl>

Entsprechend erklärt man den Begriff des Homomorphismus zwischen Linksmoduln:

Eine Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\rightarrow N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/190f9ff6f9f4acba07624a16eb6be3fc9b5a96bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\rightarrow N}" loading="lazy"> zwischen zwei Linksmoduln <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> über dem Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> heißt Homomorphismus von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><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">, wenn für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{1},m_{2}\in M}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{1},m_{2}\in M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05c2d2ccf329e110868d1a76cce2f4fc3478a0ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.506ex; height:2.509ex;" alt="{\displaystyle m_{1},m_{2}\in M}" loading="lazy"> und alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca49c66b5e9b5f32249a737e4429c3df136c33f0.svg" 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}" loading="lazy"> gilt:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(m_{1}+m_{2})=f(m_{1})+f(m_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70f2ecb34ba94f947aeaa8cd7617a3ffb9341c89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.421ex; height:2.843ex;" alt="{\displaystyle f(m_{1}+m_{2})=f(m_{1})+f(m_{2})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(r\cdot m_{1})=r\cdot f(m_{1})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(r\cdot m_{1})=r\cdot f(m_{1})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9c6db4c2324dfcb0a56c0b9b368837a556e32fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.919ex; height:2.843ex;" alt="{\displaystyle f(r\cdot m_{1})=r\cdot f(m_{1})}" loading="lazy"></dd></dl>

Die Menge der Homomorphismen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> wird mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} _{R}(M,N)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{R}(M,N)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45796f476c5d2a7d1e413ab8349e274fe16c25d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.67ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} _{R}(M,N)}" loading="lazy"> bezeichnet.

Ein Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\rightarrow M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bad1b349005bcbde8aa0f0159d0a4d33b3503c19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.811ex; height:2.509ex;" alt="{\displaystyle f\colon M\rightarrow M}" loading="lazy"> von einem Modul <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> in sich selbst heißt Endomorphismus von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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">.

Sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> zwei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><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">-<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><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">-<a href="Modul_(Mathematik)#Bimoduln" title="Modul (Mathematik)">Bimoduln</a> über Ringen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><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">, so heißt eine Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\to N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26c86d8613d6babf1a7cd6dd5909a3b480840c56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\to N}" loading="lazy"> ein Homomorphismus von S-R-Bimoduln, wenn für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m,m_{1},m_{2}\in M,\ s\in S,\ r\in R}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m,m_{1},m_{2}\in M,\ s\in S,\ r\in R}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e345297f0356515d719e62d69f582bb203d754c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.893ex; height:2.509ex;" alt="{\displaystyle m,m_{1},m_{2}\in M,\ s\in S,\ r\in R}" loading="lazy"> gilt:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(m_{1}+m_{2})=f(m_{1})+f(m_{2})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70f2ecb34ba94f947aeaa8cd7617a3ffb9341c89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.421ex; height:2.843ex;" alt="{\displaystyle f(m_{1}+m_{2})=f(m_{1})+f(m_{2})}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(s\cdot m\cdot r)=s\cdot f(m\cdot r)=s\cdot f(m)\cdot r}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(s\cdot m\cdot r)=s\cdot f(m\cdot r)=s\cdot f(m)\cdot r}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb1a6eb2cfefb48259c80568ca8bec979d0ff519.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.074ex; height:2.843ex;" alt="{\displaystyle f(s\cdot m\cdot r)=s\cdot f(m\cdot r)=s\cdot f(m)\cdot r}" loading="lazy">.</dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Beispiele">Beispiele</h3></div>

</mstyle>

</mrow>

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

</semantics>

</math><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"> ein beliebiger Modul, so gibt es genau einen Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\colon \{0\}\rightarrow M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\colon \{0\}\rightarrow M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9a16012424039dc320d82c4a929b420c437bf90.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.74ex; height:2.843ex;" alt="{\displaystyle 0\colon \{0\}\rightarrow M}" loading="lazy">, nämlich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\mapsto 0\in M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\mapsto 0\in M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1edb585d0932e345e4e092a49c3800bb46aa4c6f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.222ex; height:2.176ex;" alt="{\displaystyle 0\mapsto 0\in M}" loading="lazy">. Es ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ff0df9ef65c0572eb676580ce1c02b8ec40f694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}" loading="lazy"> ein <a href="Anfangsobjekt%2C_Endobjekt_und_Nullobjekt" title="Anfangsobjekt, Endobjekt und Nullobjekt">Anfangsobjekt</a> in der Kategorie der Rechtsmoduln. Genauso gibt es nur einen Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\to \{0\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\to \{0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61cc9a2013442d346aa2845aac26137d68bf7c04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.544ex; height:2.843ex;" alt="{\displaystyle M\to \{0\}}" loading="lazy">, die <a href="Nullabbildung" class="mw-redirect" title="Nullabbildung">Nullabbildung</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\mapsto 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\mapsto 0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/33d2e3b12fdc5c200348e47160110c1cf2f08d33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.817ex; height:2.176ex;" alt="{\displaystyle m\mapsto 0}" loading="lazy"> für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\in M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c5790f9afa538086b9fea356114e77099c1a775.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.323ex; height:2.176ex;" alt="{\displaystyle m\in M}" loading="lazy">). Es ist auch <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ff0df9ef65c0572eb676580ce1c02b8ec40f694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}" loading="lazy"> ein Endobjekt. Man fasst zusammen, wenn man sagt, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ff0df9ef65c0572eb676580ce1c02b8ec40f694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}" loading="lazy"> ist ein Nullobjekt.</li>

<li>Die Identität <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{M}\colon M\to M,\,m\mapsto m}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{M}\colon M\to M,\,m\mapsto m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91e1decf4411879aa7bb0339cf5cd8916251b1bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.944ex; height:2.509ex;" alt="{\displaystyle \mathbf {1} _{M}\colon M\to M,\,m\mapsto m}" loading="lazy"> ist ein Homomorphismus.</li>

<li>Das Zentrum eines Ringes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> ist die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=\{s\in R\mid s\cdot r=r\cdot s{\text{ für alle }}r\in R\}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z=\{s\in R\mid s\cdot r=r\cdot s{\text{ für alle }}r\in R\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb51a22215073af0f6c86bcb2599f9348fcf021e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:39.395ex; height:3.343ex;" alt="{\displaystyle Z=\{s\in R\mid s\cdot r=r\cdot s{\text{ für alle }}r\in R\}}" loading="lazy"> ist ein Unterring des Ringes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><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">. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01d131dfd7673938b947072a13a9744fe997e632.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}" loading="lazy"> im Zentrum des Ringes, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l(s)\colon M\to M,\,m\mapsto m\cdot s}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle l(s)\colon M\to M,\,m\mapsto m\cdot s}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99d569616252922c45e540def7e4d5ea593ad521.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.011ex; height:2.843ex;" alt="{\displaystyle l(s)\colon M\to M,\,m\mapsto m\cdot s}" loading="lazy"> ein Homomorphismus.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f,g\colon M\rightarrow N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/736284ed0a643330ad0accaf7ba8960ed20080a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.582ex; height:2.509ex;" alt="{\displaystyle f,g\colon M\rightarrow N}" loading="lazy"> zwei Homomorphismen, so ist ihre Summe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+g\colon M\to N,\,m\mapsto f(m)+g(m)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f+g\colon M\to N,\,m\mapsto f(m)+g(m)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f45bc1a89d4344ca4d46927c8db60d44a3374e49.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.399ex; height:2.843ex;" alt="{\displaystyle f+g\colon M\to N,\,m\mapsto f(m)+g(m)}" loading="lazy"> ein Homomorphismus.</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Eigenschaften">Eigenschaften</h3></div>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/190f9ff6f9f4acba07624a16eb6be3fc9b5a96bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\rightarrow N}" loading="lazy"> ein Homomorphismus und ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\hookrightarrow N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2150c5b596c24e178547c2b2a22fb6767f4b19c5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.758ex; height:2.176ex;" alt="{\displaystyle V\hookrightarrow N}" loading="lazy"> ein <a href="Untermodul" title="Untermodul">Untermodul</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(V):=\{m\in M\mid f(m)\in V\}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(V):=\{m\in M\mid f(m)\in V\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba9c12dc6b1aa08a2c3d5568e90b15964885a7e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.337ex; height:3.176ex;" alt="{\displaystyle f^{-1}(V):=\{m\in M\mid f(m)\in V\}}" loading="lazy"> ein Untermodul von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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">. Insbesondere ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{-1}(\{0\})}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{-1}(\{0\})}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48fdef0255c77b154755ab04e729012839fa831c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.95ex; height:3.176ex;" alt="{\displaystyle f^{-1}(\{0\})}" loading="lazy"> ein Untermodul von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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">. Dieser Untermodul heißt <a href="Kern_(Algebra)" title="Kern (Algebra)">Kern</a> des Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">. Er wird oft mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Kern} (f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Kern} (f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cdeaa6898f2903f255cefb6db1907ddc37f2bcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.133ex; height:2.843ex;" alt="{\displaystyle \operatorname {Kern} (f)}" loading="lazy"> oder auch einfach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Ke} (f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Ke} (f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b7ae24424e01427e43138a779784e8746012c98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.928ex; height:2.843ex;" alt="{\displaystyle \operatorname {Ke} (f)}" loading="lazy"> bezeichnet.</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> ein Untermodul von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\to N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/26c86d8613d6babf1a7cd6dd5909a3b480840c56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\to N}" loading="lazy"> ein Modulhomomorphismus, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(U)=\{f(u)\mid u\in U\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(U)=\{f(u)\mid u\in U\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa5eb0f0437836a8dce86a82cb63258410121fc5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.602ex; height:2.843ex;" alt="{\displaystyle f(U)=\{f(u)\mid u\in U\}}" loading="lazy"> ein Untermodul von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><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">. Er heißt Bild von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> unter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">. Insbesondere ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(M)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/838e63aefd54118a54c25fd9f0ef22528a69e36c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.53ex; height:2.843ex;" alt="{\displaystyle f(M)}" loading="lazy">, die Bildmenge von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">, ein Untermodul von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><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">. Er wird oft mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Bild} (f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Bild} (f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e46985b7f8a624df067912d78a2d1f723d5fc017.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.32ex; height:2.843ex;" alt="{\displaystyle \operatorname {Bild} (f)}" loading="lazy"> oder einfach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Bi} (f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Bi} (f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c787ce778ddeaff663ff1528ce71bc9ad808125f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.38ex; height:2.843ex;" alt="{\displaystyle \operatorname {Bi} (f)}" loading="lazy"> bezeichnet.</li>

<li>Die Verkettung oder <a href="Komposition_(Mathematik)" title="Komposition (Mathematik)">Komposition</a> zweier Homomorphismen ist ein Homomorphismus. Die Menge der Moduln über einem Ring bilden zusammen mit den Homomorphismen eine <a href="Kategorientheorie" title="Kategorientheorie">Kategorie</a>.</li>

</mstyle>

</mrow>

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

</semantics>

</math><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"> ein Modul, so bildet die Menge der <a href="Endomorphismus" title="Endomorphismus">Endomorphismen</a> einen unitären Ring. Dabei ist die Addition die Addition der Endomorphismen und die Multiplikation ist die Verkettung.</li></ul>

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

Für einen Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M_{R}\rightarrow N_{R}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon M_{R}\rightarrow N_{R}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbdee9c05ceae4a24fe6f82bcbb358c0afbd0df6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.006ex; height:2.509ex;" alt="{\displaystyle f\colon M_{R}\rightarrow N_{R}}" loading="lazy">

zwischen Moduln sind folgende Aussagen äquivalent.

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Kern} (f)=\{m\in M\mid f(m)=0\}=\{0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba9715cca6c50f02a36269886281888c313384c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.791ex; height:2.843ex;" alt="{\displaystyle \operatorname {Kern} (f)=\{m\in M\mid f(m)=0\}=\{0\}}" loading="lazy"></li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dbb00e3fe8849131890d2b4db8373457573bd0a4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.544ex; height:2.509ex;" alt="{\displaystyle a,b\in M}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(a)=f(b)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f70209bf143b8417feef2aed98b2e86bc8f447e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.502ex; height:2.843ex;" alt="{\displaystyle f(a)=f(b)}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=b}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1956b03d1314c7071ac1f45ed7b1e29422dcfcc4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="{\displaystyle a=b}" loading="lazy">.</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> ist links kürzbar. Das heißt: Für alle Moduln <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{R}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6b6f840fe97e3fe73c8e6d9c0a51f01f9054310.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.062ex; height:2.509ex;" alt="{\displaystyle L_{R}}" loading="lazy"> und alle Homomorphismen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g,h:L_{R}\rightarrow M_{R}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g,h:L_{R}\rightarrow M_{R}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18166ae7e5699bfb675d544d6a12fcab0c147d54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.836ex; height:2.509ex;" alt="{\displaystyle g,h:L_{R}\rightarrow M_{R}}" loading="lazy"> gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle fg=fh\Rightarrow g=h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle fg=fh\Rightarrow g=h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a95a81d953bca8177075dd33ebd3c4dbbd05f246.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.278ex; height:2.509ex;" alt="{\displaystyle fg=fh\Rightarrow g=h}" loading="lazy">.</li></ol>

Erfüllt ein Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M_{R}\rightarrow N_{R}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\colon M_{R}\rightarrow N_{R}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbdee9c05ceae4a24fe6f82bcbb358c0afbd0df6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.006ex; height:2.509ex;" alt="{\displaystyle f\colon M_{R}\rightarrow N_{R}}" loading="lazy"> eine und damit alle äquivalenten Eigenschaften des Satzes, so heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\,}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba5397b34cab7d96daac496a937b0c0fa076dff7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.666ex; height:2.509ex;" alt="{\displaystyle f\,}" loading="lazy"> Monomorphismus zwischen den Moduln. Die dritte Aussage des Satzes besagt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> im Sinne der <a href="Kategorientheorie" title="Kategorientheorie">Kategorientheorie</a> ein <a href="Monomorphismus" title="Monomorphismus">Monomorphismus</a> ist.

<div class="mw-heading mw-heading3"><h3 id="Beispiele_2">Beispiele</h3></div>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82056ed7abb71b6d871992e72e22773a812ac64c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.132ex; height:2.176ex;" alt="{\displaystyle U\hookrightarrow M}" loading="lazy"> ein <a href="Untermodul" title="Untermodul">Untermodul</a>, so ist die Inklusionsabbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \iota \colon U\to M,\,u\mapsto u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \iota \colon U\to M,\,u\mapsto u}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/dc83cd44011284320c266af7205452bb7712f346.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.391ex; height:2.509ex;" alt="{\displaystyle \iota \colon U\to M,\,u\mapsto u}" loading="lazy"> ein Monomorphismus.</li>

<li>Jeder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy">-Homomorphismus von der Menge der ganzen Zahlen in die rationalen Zahlen, welcher nicht die Nullabbildung ist, ist ein Monomorphismus.</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Bemerkungen">Bemerkungen</h3></div>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/190f9ff6f9f4acba07624a16eb6be3fc9b5a96bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\rightarrow N}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon N\rightarrow O}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6845b6439c2881ea7d659a9c7b6ebacfe5d268cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.601ex; height:2.509ex;" alt="{\displaystyle g\colon N\rightarrow O}" loading="lazy"> Monomorphismen, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}" loading="lazy"> ein Monomorphismus.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}" loading="lazy"> ein Monomorphismus, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> ein Monomorphismus.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\circ f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/10b5ad4985af48d0fb7efa3c8afa5ad7d42bfc92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g\circ f}" loading="lazy"> ein Monomorphismus, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(M)\cap \operatorname {Kern} (g)=\{0\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(M)\cap \operatorname {Kern} (g)=\{0\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f17106e41f8c34bbf9b7f59b1b5d5b653ad1c9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.668ex; height:2.843ex;" alt="{\displaystyle f(M)\cap \operatorname {Kern} (g)=\{0\}}" loading="lazy">.</li></ol>

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

<div class="mw-heading mw-heading3"><h3 id="Definition_2">Definition</h3></div>

Für einen Modulhomomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in \operatorname {Hom} _{R}(M,N)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in \operatorname {Hom} _{R}(M,N)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0054c34a98746b66ecbde06e5b36fe89b432023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.789ex; height:2.843ex;" alt="{\displaystyle f\in \operatorname {Hom} _{R}(M,N)}" loading="lazy"> sind folgende Aussagen äquivalent:

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N/f(M)=0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a9fac55a484cd8f00915f7158de6431314ad1d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.017ex; height:2.843ex;" alt="{\displaystyle N/f(M)=0}" loading="lazy">. Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N/f(M)}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a6e7e7f97298e0bb29a1657abf2591521d7ba3f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.756ex; height:2.843ex;" alt="{\displaystyle N/f(M)}" loading="lazy"> der <a href="Quotientenmodul" title="Quotientenmodul">Faktormodul von N modulo f(M)</a>.</li>

<li>Die Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> ist <a href="Surjektivit%C3%A4t" class="mw-redirect" title="Surjektivität">surjektiv</a>.</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> ist rechts kürzbar. Das heißt, für alle Moduln <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> und alle Homomorphismen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g,h\in \operatorname {Hom} _{R}(N,P)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g,h\in \operatorname {Hom} _{R}(N,P)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ab6c890293771502fad44513c9cbb7760336bea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.303ex; height:2.843ex;" alt="{\displaystyle g,h\in \operatorname {Hom} _{R}(N,P)}" loading="lazy"> gilt: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gf=hf\Rightarrow g=h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle gf=hf\Rightarrow g=h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/797a5765bf75c657d628be58b1d954712c2e74a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.278ex; height:2.509ex;" alt="{\displaystyle gf=hf\Rightarrow g=h}" loading="lazy">.</li></ol>

Ein Homomorphismus, der eine und damit alle diese Eigenschaften erfüllt, heißt Epimorphismus. Die dritte Eigenschaft des Satzes besagt, dass der Homomorphismus im Sinne der <a href="Kategorientheorie" title="Kategorientheorie">Kategorientheorie</a> ein <a href="Epimorphismus" title="Epimorphismus">Epimorphismus</a> ist.

<div class="mw-heading mw-heading3"><h3 id="Beispiele_3">Beispiele</h3></div>

<ol><li>Die Identität <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\to M,\,m\mapsto m}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffb8bccb2788347f768bc3b7a1b5b4cd72a630b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.614ex; height:2.509ex;" alt="{\displaystyle M\to M,\,m\mapsto m}" loading="lazy"> ist ein Epimorphismus.</li>

</mstyle>

</mrow>

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

</semantics>

</math><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"> ein <a href="Integrit%C3%A4tsring" title="Integritätsring">Integritätsring</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> sein <a href="Quotientenk%C3%B6rper" title="Quotientenkörper">Quotientenkörper</a>, so ist jeder Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\neq f\in \operatorname {Hom} _{R}(K,K)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\neq f\in \operatorname {Hom} _{R}(K,K)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02ecc782e4d560529602633eb27e51153586374c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.676ex; height:2.843ex;" alt="{\displaystyle 0\neq f\in \operatorname {Hom} _{R}(K,K)}" loading="lazy"> ein Monomorphismus und ein Epimorphismus.</li>

<li>Es sei p eine Primzahl und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} [p^{-1}]=\{z\cdot p^{-i}\mid z\in \mathbb {Z} ,i\in \mathbb {N} \}}">

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [p^{-1}]=\{z\cdot p^{-i}\mid z\in \mathbb {Z} ,i\in \mathbb {N} \}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a52fc28b55ceba2adc17e8d9dfc2872498dacc14.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.556ex; height:3.176ex;" alt="{\displaystyle \mathbb {Z} [p^{-1}]=\{z\cdot p^{-i}\mid z\in \mathbb {Z} ,i\in \mathbb {N} \}}" loading="lazy"> der kleinste Unterring der rationalen Zahlen, der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47069989615a19088a9c08f5a37c8d95426447e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.592ex; height:3.009ex;" alt="{\displaystyle p^{-1}}" loading="lazy"> enthält. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathbb {Z} }:=\mathbb {Z} [p^{-1}]/\mathbb {Z} }">

<msub>

</mrow>

</msub>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{\mathbb {Z} }:=\mathbb {Z} [p^{-1}]/\mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/91b9a0bda1580d7909462aa3e8ec0b427abbd61a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.387ex; height:3.176ex;" alt="{\displaystyle M_{\mathbb {Z} }:=\mathbb {Z} [p^{-1}]/\mathbb {Z} }" loading="lazy">, so ist jeder Endomorphismus von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{\mathbb {Z} }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M_{\mathbb {Z} }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b4fb3e43a50402069d90aa7c3170dc0c9a8caa6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.583ex; height:2.509ex;" alt="{\displaystyle M_{\mathbb {Z} }}" loading="lazy">, der ungleich der Nullabbildung ist, ein Epimorphismus. Aber die Multiplikation mit p ist kein Monomorphismus.</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Eigenschaften_2">Eigenschaften</h3></div>

<ol><li>Die Verkettung von Epimorphismen ist ein Epimorphismus.</li>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in \operatorname {Hom} _{R}(M,N),\quad g\in \operatorname {Hom} _{R}(N,P)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9a716e988d4e025dfc4d5d6000ef73f6e2ca997.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.076ex; height:2.843ex;" alt="{\displaystyle f\in \operatorname {Hom} _{R}(M,N),\quad g\in \operatorname {Hom} _{R}(N,P)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle gf}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ae6992a2a836c1ff200f058911a5a15f32de24c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.395ex; height:2.509ex;" alt="{\displaystyle gf}" loading="lazy"> ein Epimorphismus, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> ein Epimorphismus und es ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Kern} (g)+f(M)=N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Kern} (g)+f(M)=N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd524c42be4f45879f412156010f1db11e11618d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.503ex; height:2.843ex;" alt="{\displaystyle \operatorname {Kern} (g)+f(M)=N}" loading="lazy">.</li></ol>

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

Ein Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\rightarrow N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/190f9ff6f9f4acba07624a16eb6be3fc9b5a96bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\rightarrow N}" loading="lazy"> heißt Isomorphismus, wenn es einen Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon N\rightarrow M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9863dac9c474667635c2b4a58a2a0b5c19559ad9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.27ex; height:2.509ex;" alt="{\displaystyle g\colon N\rightarrow M}" loading="lazy"> gibt, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\circ f=\mathbf {1} _{M}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g\circ f=\mathbf {1} _{M}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/69383509ab1d2d33c994f1cf28e6ac584170b209.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.984ex; height:2.509ex;" alt="{\displaystyle g\circ f=\mathbf {1} _{M}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g=\mathbf {1} _{N}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\circ g=\mathbf {1} _{N}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d40569f9351f41e2cd4091c20ad6e58bfeb1856.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.716ex; height:2.509ex;" alt="{\displaystyle f\circ g=\mathbf {1} _{N}}" loading="lazy"> ist. Dies ist genau dann der Fall, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> ein Monomorphismus und ein Epimorphismus ist. Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{M}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{M}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0447b4c2d9088a823983a08e2db928ab38917dc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.296ex; height:2.509ex;" alt="{\displaystyle \mathbf {1} _{M}}" loading="lazy"> die Identität auf dem Modul <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {1} _{N}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {1} _{N}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92405b45fa8bf31efbfe9d79014d37ab768fa21d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.028ex; height:2.509ex;" alt="{\displaystyle \mathbf {1} _{N}}" loading="lazy"> analog die Identität auf dem Modul <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><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">. Zwei Moduln <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30012178e674e69a55fee64e9c221dbdb0d95b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle M,N}" loading="lazy"> heißen isomorph, in Zeichen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\cong N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f85d0d3e78a1f157d37873c13b05e150eb43aba3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.604ex; height:2.176ex;" alt="{\displaystyle M\cong N}" loading="lazy">, wenn es einen Isomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha :M\rightarrow N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb87f0cfb40e4081425b2968f4b24c8b9437bd58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.545ex; height:2.176ex;" alt="{\displaystyle \alpha :M\rightarrow N}" loading="lazy"> gibt.

<div class="mw-heading mw-heading2"><h2 id="Produktzerlegungen_von_Homomorphismen">Produktzerlegungen von Homomorphismen</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Homomorphiesatz">Homomorphiesatz</h3></div>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\rightarrow N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/190f9ff6f9f4acba07624a16eb6be3fc9b5a96bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\rightarrow N}" loading="lazy"> ein Homomorphismus, so gibt es einen eindeutig bestimmten Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}\colon M/\operatorname {Kern} (f)\to N}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{*}\colon M/\operatorname {Kern} (f)\to N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b76950ffa98aceb4a8f5387e572b893a7ee72d28.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.211ex; height:2.843ex;" alt="{\displaystyle f^{*}\colon M/\operatorname {Kern} (f)\to N}" loading="lazy">, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}\pi =f}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{*}\pi =f}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5be07aa963e041695e60bb4a60c852ab7363de97.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.084ex; height:2.676ex;" alt="{\displaystyle f^{*}\pi =f}" loading="lazy"> gilt. Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi \colon M\to M/\operatorname {Kern} (f)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi \colon M\to M/\operatorname {Kern} (f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5886bee566b3e31a3759f3721fea02fc629c1099.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.547ex; height:2.843ex;" alt="{\displaystyle \pi \colon M\to M/\operatorname {Kern} (f)}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\mapsto m+\operatorname {Kern} (f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\mapsto m+\operatorname {Kern} (f)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/262c2eefddfabad6143f4069ccf531e0407a1e40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.668ex; height:2.843ex;" alt="{\displaystyle m\mapsto m+\operatorname {Kern} (f)}" loading="lazy"> der kanonische Epimorphismus. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{*}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/190a73fde235865b8d2a783334f90194331c7f19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.375ex; height:2.676ex;" alt="{\displaystyle f^{*}}" loading="lazy"> ist stets ein Monomorphismus. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> ein Epimorphismus, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{*}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/190a73fde235865b8d2a783334f90194331c7f19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.375ex; height:2.676ex;" alt="{\displaystyle f^{*}}" loading="lazy"> ein Isomorphismus.

Der <a href="Homomorphiesatz" title="Homomorphiesatz">Homomorphiesatz</a> besagt also, dass das folgende <a href="Kommutatives_Diagramm" title="Kommutatives Diagramm">Diagramm kommutiert</a>.

<mtr>

<mtd>

</mtd>

<mtd>

<mover>

</mrow>

</mrow>

</mover>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<msub>

</mrow>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</mtd>

<mtd>

<mover>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mover>

</mrow>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{array}{ccl}M&{\stackrel {f}{\longrightarrow }}&N\\\pi {\Big \downarrow }&&\parallel \mathbf {1} _{N}\\M/\operatorname {Kern} (f)&{\stackrel {f^{*}}{\longrightarrow }}&N\end{array}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e50313903812882059d8ce6dec6fad50a260aacf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:25.517ex; height:13.509ex;" alt="{\displaystyle {\begin{array}{ccl}M&{\stackrel {f}{\longrightarrow }}&N\\\pi {\Big \downarrow }&&\parallel \mathbf {1} _{N}\\M/\operatorname {Kern} (f)&{\stackrel {f^{*}}{\longrightarrow }}&N\end{array}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="1._Isomorphiesatz">1. Isomorphiesatz</h3></div>

Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U,V\hookrightarrow M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/249c541bc4b8ba1800da7407308ace0b34e8edb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.953ex; height:2.509ex;" alt="{\displaystyle U,V\hookrightarrow M}" loading="lazy"> Untermoduln von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> Dann gilt:

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed8a274024ae4b9b6549c2b96965e8ac574bc908.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.174ex; height:2.843ex;" alt="{\displaystyle (U+V)/V\cong U/(U\cap V)}" loading="lazy">. Der Isomorphismus ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U+V)/V\ni u+V\mapsto u+U\cap V\in U/(U\cap V)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (U+V)/V\ni u+V\mapsto u+U\cap V\in U/(U\cap V)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eabf376cd83b24227186a10bdf2383244fd676fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.651ex; height:2.843ex;" alt="{\displaystyle (U+V)/V\ni u+V\mapsto u+U\cap V\in U/(U\cap V)}" loading="lazy">

Folgerung:

Seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\hookrightarrow M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82056ed7abb71b6d871992e72e22773a812ac64c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.132ex; height:2.176ex;" alt="{\displaystyle U\hookrightarrow M}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\hookrightarrow M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/12361f91f2f7f5717ee1ba86975b9dacad5c2178.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.136ex; height:2.176ex;" alt="{\displaystyle V\hookrightarrow M}" loading="lazy"> Untermoduln von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\oplus V=M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/504d41d83d6a72d15b2c4ac99d442b3f58410411.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:11.951ex; height:2.343ex;" alt="{\displaystyle U\oplus V=M}" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\cong M/V}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5323ff85a250f416368ea592bee4e31813151a7d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.273ex; height:2.843ex;" alt="{\displaystyle U\cong M/V}" loading="lazy">.<a href="#cite_note-1">[1]</a>

<div class="mw-heading mw-heading3"><h3 id="2._Isomorphiesatz">2. Isomorphiesatz</h3></div>

Es seien <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U\hookrightarrow V\hookrightarrow M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9009921ab1c342b04a2f529e025f3325879bfec6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.825ex; height:2.176ex;" alt="{\displaystyle U\hookrightarrow V\hookrightarrow M}" loading="lazy"> Untermoduln von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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">. Dann gilt:

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M/V\cong (M/U)/(V/U)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9837afa56460bed04013c8f5bb10d5310092546c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.391ex; height:2.843ex;" alt="{\displaystyle M/V\cong (M/U)/(V/U)}" loading="lazy">.<a href="#cite_note-2">[2]</a></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Der_Hom-Funktor">Der Hom-Funktor</h2></div>

Sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M,N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30012178e674e69a55fee64e9c221dbdb0d95b9e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle M,N}" loading="lazy"> Moduln, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (M,N)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (M,N)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b3a724afd4d80438861c7c21fa8e14180a4b2cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.19ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (M,N)}" loading="lazy"> die Menge der Homomorphismen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\rightarrow N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/190f9ff6f9f4acba07624a16eb6be3fc9b5a96bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.432ex; height:2.509ex;" alt="{\displaystyle f\colon M\rightarrow N}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Moduleigenschaften_von_Hom">Moduleigenschaften von Hom</h3></div>

<ul><li>Die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (M,N)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (M,N)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b3a724afd4d80438861c7c21fa8e14180a4b2cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.19ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (M,N)}" loading="lazy"> wird zu einer abelschen Gruppe, wenn für zwei Homomorphismen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g\colon M\rightarrow N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f,g\colon M\rightarrow N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/736284ed0a643330ad0accaf7ba8960ed20080a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.582ex; height:2.509ex;" alt="{\displaystyle f,g\colon M\rightarrow N}" loading="lazy"> die Summe folgendermaßen definiert ist: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f+g\colon M\ni m\mapsto f(m)+g(m)\in N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f+g\colon M\ni m\mapsto f(m)+g(m)\in N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b12fb257d84ad9cde7ea636dca967e404a5af39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.045ex; height:2.843ex;" alt="{\displaystyle f+g\colon M\ni m\mapsto f(m)+g(m)\in N}" loading="lazy">.</li>

</mstyle>

</mrow>

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

</semantics>

</math><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"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S-R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6b3a0a1bf62e8adb590bd771cf297289471389a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.104ex; height:2.343ex;" alt="{\displaystyle S-R}" loading="lazy"> Bimodul, auf der linken Seite ein Modul über dem Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> und auf der rechten Seite ein Modul über dem Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><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">, so wird <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} _{R}(M,N)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{R}(M,N)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45796f476c5d2a7d1e413ab8349e274fe16c25d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.67ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} _{R}(M,N)}" loading="lazy"> auf der rechten Seite zu einem Modul über dem Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><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">, wenn man für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in \operatorname {Hom} _{R}(M,N)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in \operatorname {Hom} _{R}(M,N)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0054c34a98746b66ecbde06e5b36fe89b432023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.789ex; height:2.843ex;" alt="{\displaystyle f\in \operatorname {Hom} _{R}(M,N)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acce52dffd84d073a24f4606a175da60148fd0c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.43ex; height:2.176ex;" alt="{\displaystyle s\in S}" loading="lazy"> definiert: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\cdot s\colon M\ni m\mapsto f(s\cdot m)\in N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\cdot s\colon M\ni m\mapsto f(s\cdot m)\in N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac8658ce756dd858299b03c653dfeea6329fbd33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.822ex; height:2.843ex;" alt="{\displaystyle f\cdot s\colon M\ni m\mapsto f(s\cdot m)\in N}" loading="lazy">. Ist insbesondere <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> der Endomorphismenring von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (M,N)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (M,N)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b3a724afd4d80438861c7c21fa8e14180a4b2cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.19ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (M,N)}" loading="lazy"> auf der rechten Seite ein Modul über dem Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><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">.</li>

</mstyle>

</mrow>

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

</semantics>

</math><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"> ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S-R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6b3a0a1bf62e8adb590bd771cf297289471389a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.104ex; height:2.343ex;" alt="{\displaystyle S-R}" loading="lazy"> Bimodul, auf der linken Seite ein Modul über dem Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> und auf der rechten Seite über dem Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><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">, so wird <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} _{R}(M,N)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} _{R}(M,N)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/45796f476c5d2a7d1e413ab8349e274fe16c25d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.67ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} _{R}(M,N)}" loading="lazy"> auf der linken Seite zu einem Modul über dem Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><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">, wenn man für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in \operatorname {Hom} _{R}(M,N)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\in \operatorname {Hom} _{R}(M,N)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a0054c34a98746b66ecbde06e5b36fe89b432023.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.789ex; height:2.843ex;" alt="{\displaystyle f\in \operatorname {Hom} _{R}(M,N)}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acce52dffd84d073a24f4606a175da60148fd0c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.43ex; height:2.176ex;" alt="{\displaystyle s\in S}" loading="lazy"> definiert: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\cdot f\colon M\ni m\mapsto s\cdot f(m)\in N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\cdot f\colon M\ni m\mapsto s\cdot f(m)\in N}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3573f1579430649ba096e26fd8889c7688b60fec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.822ex; height:2.843ex;" alt="{\displaystyle s\cdot f\colon M\ni m\mapsto s\cdot f(m)\in N}" loading="lazy">.</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Der_kovariante_Funktor_Hom">Der kovariante Funktor Hom</h3></div>

Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> ein Modul, so ordnet man jedem Modul <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> die abelsche Gruppe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (M,N)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (M,N)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b3a724afd4d80438861c7c21fa8e14180a4b2cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.19ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (M,N)}" loading="lazy"> zu. Jedem Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \colon A\rightarrow B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \colon A\rightarrow B}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e07293e4f2ecbb3259ba739a0ba239bc9129ce2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.643ex; height:2.176ex;" alt="{\displaystyle \alpha \colon A\rightarrow B}" loading="lazy"> wird der Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (M,\alpha )\colon \operatorname {Hom} (M,A)\ni f\mapsto \alpha \circ f\in \operatorname {Hom} (M,B)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (M,\alpha )\colon \operatorname {Hom} (M,A)\ni f\mapsto \alpha \circ f\in \operatorname {Hom} (M,B)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b103e7b36af705cb5110592f572cb2c11f6c2a56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:51.944ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (M,\alpha )\colon \operatorname {Hom} (M,A)\ni f\mapsto \alpha \circ f\in \operatorname {Hom} (M,B)}" loading="lazy"> zugeordnet. Es gilt dann für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \colon A\rightarrow B\,\,,\beta \colon B\rightarrow C}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \colon A\rightarrow B\,\,,\beta \colon B\rightarrow C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ce4286c390e6643ec830ae6e7c7af859ed20d596.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.961ex; height:2.509ex;" alt="{\displaystyle \alpha \colon A\rightarrow B\,\,,\beta \colon B\rightarrow C}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (M,\beta \circ \alpha )=\operatorname {Hom} (M,\alpha )\circ \operatorname {Hom} (M,\beta )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (M,\beta \circ \alpha )=\operatorname {Hom} (M,\alpha )\circ \operatorname {Hom} (M,\beta )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d241c7c6bcdcaafb82c5f4ba0ec858053ed07e12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.508ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (M,\beta \circ \alpha )=\operatorname {Hom} (M,\alpha )\circ \operatorname {Hom} (M,\beta )}" loading="lazy">. Außerdem werden die Identitäten auf die entsprechenden Identitäten abgebildet. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (M,-)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (M,-)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00c4b90fdf17c2a128b60a3cf66da2cada85b8d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.935ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (M,-)}" loading="lazy"> ist ein <a href="Kovarianter_Funktor" class="mw-redirect" title="Kovarianter Funktor">kovarianter Funktor</a> von der Kategorie der Moduln über dem Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> in die Kategorie der abelschen Gruppen. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> wie oben ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S-R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c6b3a0a1bf62e8adb590bd771cf297289471389a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.104ex; height:2.343ex;" alt="{\displaystyle S-R}" loading="lazy"> Bimodul, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (M,-)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (M,-)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00c4b90fdf17c2a128b60a3cf66da2cada85b8d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.935ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (M,-)}" loading="lazy"> ein Funktor von der Kategorie der Moduln über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> in die Kategorie der Moduln über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

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

</semantics>

</math><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">.

<div class="mw-heading mw-heading3"><h3 id="Linksexaktheit_von_Hom">Linksexaktheit von Hom</h3></div>

Für einen Komplex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\rightarrow A\;{\overset {\alpha }{\rightarrow }}\;B\;{\overset {\beta }{\rightarrow }}\;C}">

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\rightarrow A\;{\overset {\alpha }{\rightarrow }}\;B\;{\overset {\beta }{\rightarrow }}\;C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa8559da641dd7afc0515c0cb906bc97461f9003.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.278ex; height:3.676ex;" alt="{\displaystyle 0\rightarrow A\;{\overset {\alpha }{\rightarrow }}\;B\;{\overset {\beta }{\rightarrow }}\;C}" loading="lazy">, das heißt, es gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \circ \alpha =0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \beta \circ \alpha =0}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/151de91da8b3870d9d0440b125c1fe3cc0460b8c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.275ex; height:2.509ex;" alt="{\displaystyle \beta \circ \alpha =0}" loading="lazy">, sind die folgenden Aussagen äquivalent:

<mover>

</mover>

</mrow>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\rightarrow A\;{\overset {\alpha }{\rightarrow }}\;B\;{\overset {\beta }{\rightarrow }}\;C}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa8559da641dd7afc0515c0cb906bc97461f9003.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.278ex; height:3.676ex;" alt="{\displaystyle 0\rightarrow A\;{\overset {\alpha }{\rightarrow }}\;B\;{\overset {\beta }{\rightarrow }}\;C}" loading="lazy"> ist exakt.</li>

<li>Für alle Moduln <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

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

</semantics>

</math><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"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\rightarrow \operatorname {Hom} (M,A)\rightarrow \operatorname {Hom} (M,B)\rightarrow \operatorname {Hom} (M,C)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\rightarrow \operatorname {Hom} (M,A)\rightarrow \operatorname {Hom} (M,B)\rightarrow \operatorname {Hom} (M,C)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ea78f79020e551b4434656d720fcd43e1642e27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.659ex; height:2.843ex;" alt="{\displaystyle 0\rightarrow \operatorname {Hom} (M,A)\rightarrow \operatorname {Hom} (M,B)\rightarrow \operatorname {Hom} (M,C)}" loading="lazy"> <a href="Exakte_Sequenz" title="Exakte Sequenz">exakt</a>.</li>

<li>Es gibt einen <a href="Produkt_von_Moduln#Bezeichnungen_und_Beispiele" title="Produkt von Moduln">Generator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

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

</semantics>

</math><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">, so dass die Folge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\rightarrow \operatorname {Hom} (G,A)\rightarrow \operatorname {Hom} (G,B)\rightarrow \operatorname {Hom} (G,C)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\rightarrow \operatorname {Hom} (G,A)\rightarrow \operatorname {Hom} (G,B)\rightarrow \operatorname {Hom} (G,C)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fcbda3e4ddb704e6c5015137ebf86d034a818bd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.812ex; height:2.843ex;" alt="{\displaystyle 0\rightarrow \operatorname {Hom} (G,A)\rightarrow \operatorname {Hom} (G,B)\rightarrow \operatorname {Hom} (G,C)}" loading="lazy"> exakt ist.</li></ul>

Auch wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ed48a5e36207156fb792fa79d29925d2f7901e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }" loading="lazy"> surjektiv ist, so ist das für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (M,B)\rightarrow \operatorname {Hom} (M,C)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (M,B)\rightarrow \operatorname {Hom} (M,C)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/523ae678ed96711b49945ca1f145751ff4a66226.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.398ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (M,B)\rightarrow \operatorname {Hom} (M,C)}" loading="lazy"> im Allgemeinen nicht der Fall, das heißt, der Hom-Funktor ist im Allgemeinen nicht exakt. Die Abweichung von der Exaktheit wird durch den <a href="Ext_(Mathematik)" title="Ext (Mathematik)">Ext-Funktor</a> gemessen.

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Friedrich Kasch, Moduln und Ringe, Teubner, Stuttgart 1977, Seite 57

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Friedrich Kasch, Moduln und Ringe, Teubner, Stuttgart 1977, Seite 58

</li>

</ol>

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

<ul><li>Frank W. Anderson and Kent R. Fuller: Rings and Categories of Modules. Springer, New-York 1992, ISBN 0-387-97845-3</li>

<li>Friedrich Kasch: Moduln und Ringe. Teubner, Stuttgart 1977, ISBN 3-519-02211-7</li>

<li>Robert Wisbauer: Grundlagen der Modul- und Ringtheorie. Reinhard Fischer, München 1988, ISBN 3-88927-044-1</li></ul></div><div><div class="zim-footer">

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