GermanWikipedia - /page/entry.mu

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<title>Freier Modul</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Freier Modul</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">Im <a href="Mathematik" title="Mathematik">mathematischen</a> Teilgebiet der <a href="Algebra" title="Algebra">Algebra</a> ist ein freier Modul ein <a href="Modul_(Mathematik)" title="Modul (Mathematik)">Modul</a>, der eine <a href="Basis_(Modul)" title="Basis (Modul)">Basis</a> besitzt. Damit ist der Begriff des freien Moduls eine Verallgemeinerung der Begriffe <a href="Vektorraum" title="Vektorraum">Vektorraum</a> oder <a href="Freie_abelsche_Gruppe" title="Freie abelsche Gruppe">freie abelsche Gruppe</a>.

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

Eine Familie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B:=\{b_{i}\mid i\in I\}}">

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<annotation encoding="application/x-tex">{\displaystyle B:=\{b_{i}\mid i\in I\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a45455dbdbc28c1c8dd3accebf8dcbe44b662f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.383ex; height:2.843ex;" alt="{\displaystyle B:=\{b_{i}\mid i\in I\}}" loading="lazy"> von Elementen eines Moduls (oder allgemeiner eines Linksmoduls) <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_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" 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"> heißt linear unabhängig oder frei, wenn für jede endliche Indexmenge

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<annotation encoding="application/x-tex">{\displaystyle \textstyle J\subseteq I}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8192098b7899026fa25affc175b9020e5dea4469.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.742ex; height:2.343ex;" alt="{\displaystyle \textstyle J\subseteq I}" loading="lazy"> und alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle r_{i}\in R}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf61a9140f557e61a6a6f6f429cd9662995321b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.453ex; height:2.509ex;" alt="{\displaystyle \textstyle r_{i}\in R}" loading="lazy"> gilt:

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<annotation encoding="application/x-tex">{\displaystyle \sum _{i\in J}r_{i}\cdot b_{i}=0\;\Rightarrow \;\forall i\in J\colon \,r_{i}=0.}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4ce174a23d7037e6ada714dfc7b09bb45f9be0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:32.816ex; height:5.676ex;" alt="{\displaystyle \sum _{i\in J}r_{i}\cdot b_{i}=0\;\Rightarrow \;\forall i\in J\colon \,r_{i}=0.}" loading="lazy"></dd></dl>

Erzeugen die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{i}\mid i\in I\}}">

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<annotation encoding="application/x-tex">{\displaystyle \{b_{i}\mid i\in I\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6065fdf971e7ac185717d37acc9f8dfbbf7eb8d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.874ex; height:2.843ex;" alt="{\displaystyle \{b_{i}\mid i\in I\}}" loading="lazy"> zugleich den Modul <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_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">, so heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.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 B}" loading="lazy"> eine Basis (von <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_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy">) und der Modul <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_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> heißt der freie <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">-Modul über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47136aad860d145f75f3eed3022df827cee94d7a.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 B}" loading="lazy"> oder auch einfach frei.

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

<div class="mw-heading mw-heading3"><h3 id="Erste_Beispiele_und_Gegenbeispiele">Erste Beispiele und Gegenbeispiele</h3></div>

<ol><li>Jeder 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"> mit Einselement ist über sich selbst frei. Das heißt, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{R}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ed875207d483cb3387f97cd8add37a9720d9b317.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.244ex; height:2.509ex;" alt="{\displaystyle R_{R}}" loading="lazy"> ist freier Rechtsmodul. Entsprechend ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle _{R}R}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea65b6964bf5da18f540b62b81b329366124af01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.244ex; height:2.509ex;" alt="{\displaystyle _{R}R}" loading="lazy"> ein freier Linksmodul.</li>

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<annotation encoding="application/x-tex">{\displaystyle 1<n\in \mathbb {N} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/05ff214e097be2c25fde943e81c8fe96a6596c9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.174ex; height:2.176ex;" alt="{\displaystyle 1<n\in \mathbb {N} }" loading="lazy">, so ist der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }">

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

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2120ebbc85f91df66c6de5446367bf9fd620844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.658ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} /n\mathbb {Z} }" loading="lazy"> nicht frei. Der Grund ist, dass für jede <a href="Restklasse" title="Restklasse">Restklasse</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b+n\mathbb {Z} \in \mathbb {Z} /n\mathbb {Z} }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48ddd8bbf77df87497e6e5c7554295dbbb21c00c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.281ex; height:2.843ex;" alt="{\displaystyle b+n\mathbb {Z} \in \mathbb {Z} /n\mathbb {Z} }" loading="lazy"> gilt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\cdot (b+n\mathbb {Z} )=0+n\mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle n\cdot (b+n\mathbb {Z} )=0+n\mathbb {Z} }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/50573f1fc463cbee4f39a69327a0977107fe5c3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.712ex; height:2.843ex;" alt="{\displaystyle n\cdot (b+n\mathbb {Z} )=0+n\mathbb {Z} }" loading="lazy">. Keine Menge von Restklassen kann also linear unabhängig sein.</li>

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

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5909f0b54e4718fa24d5fd34d54189d24a66e9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.808ex; height:2.509ex;" alt="{\displaystyle \mathbb {Q} }" loading="lazy"> ist <a href="Torsionsfrei" class="mw-redirect" title="Torsionsfrei">torsionsfrei</a>, aber nicht frei (freie Moduln sind immer torsionsfrei).</li>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> eine natürliche Zahl, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{n}=\left\{{\begin{pmatrix}r_{1},\dots ,r_{n}\end{pmatrix}}\mid r_{1},\dots r_{n}\in R\right\}}">

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<annotation encoding="application/x-tex">{\displaystyle R^{n}=\left\{{\begin{pmatrix}r_{1},\dots ,r_{n}\end{pmatrix}}\mid r_{1},\dots r_{n}\in R\right\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a00d57a927e86d4f950a83a2fa1c35c3fd6f1e0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.571ex; height:2.843ex;" alt="{\displaystyle R^{n}=\left\{{\begin{pmatrix}r_{1},\dots ,r_{n}\end{pmatrix}}\mid r_{1},\dots r_{n}\in R\right\}}" loading="lazy"> ein freier Modul. Eine Basis ist die Familie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e_{i}\mid i\in \{1,\dots ,n\})}">

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<annotation encoding="application/x-tex">{\displaystyle (e_{i}\mid i\in \{1,\dots ,n\})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/98233e40c335a541f05188e92c3f1a4dc6b8712d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.333ex; height:2.843ex;" alt="{\displaystyle (e_{i}\mid i\in \{1,\dots ,n\})}" loading="lazy">. Dabei ist die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">-te Komponente von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebdc3a9cb1583d3204eff8918b558c293e0d2cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}" loading="lazy"> gleich <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy">, alle anderen Komponenten sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy">. Dieses Beispiel ordnet sich folgender Situation unter: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"> eine beliebige Menge, und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (F_{i}|i\in I)}">

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<annotation encoding="application/x-tex">{\displaystyle (F_{i}|i\in I)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ceda19ca6793cd5a9a2ab421a2d9a8344f8205da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.565ex; height:2.843ex;" alt="{\displaystyle (F_{i}|i\in I)}" loading="lazy"> eine Familie von Moduln, so ist das <a href="Produkt_von_Moduln#Koprodukt_von_Moduln" title="Produkt von Moduln">Koprodukt</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \bigoplus _{i\in I}F_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle \bigoplus _{i\in I}F_{i}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545963484e23f67b24251f95d94d28d6c4c5e8e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:6.192ex; height:5.676ex;" alt="{\displaystyle \bigoplus _{i\in I}F_{i}}" loading="lazy"> genau dann frei, wenn alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{i}}">

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18f9a5ed62f131c8ad9acdf7f8539f103f1e0ec9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.294ex; height:2.509ex;" alt="{\displaystyle F_{i}}" loading="lazy"> frei sind. Insbesondere ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{(I)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08c07dd98de2b46b8ccc59923f9012d98f255fcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.104ex; height:2.843ex;" alt="{\displaystyle R^{(I)}}" loading="lazy"> frei.</li>

<li>Das Produkt einer Familie von freien Moduln ist im Allgemeinen nicht frei. So ist beispielsweise <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} ^{\mathbb {N} }}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/82bd7cbe0951de092dc4a46c4b442b0c5d9efd70.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.969ex; height:2.676ex;" alt="{\displaystyle \mathbb {Z} ^{\mathbb {N} }}" loading="lazy"> nicht frei.<a href="#cite_note-1">[1]</a></li>

<li>Der <a href="Polynomring" title="Polynomring">Polynomring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle R[X]}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc1f235a6a7bf61d4f3618215d450a36191499eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.038ex; height:2.843ex;" alt="{\displaystyle \textstyle R[X]}" 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"> ist ein freier Modul mit Basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (X^{i}|i\in \mathbb {N} )}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (X^{i}|i\in \mathbb {N} )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9166166842871145da29ea02b27eb8906d0c1a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.574ex; height:3.176ex;" alt="{\displaystyle (X^{i}|i\in \mathbb {N} )}" loading="lazy">.</li>

<li>Die Menge der positiven rationalen Zahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ^{+}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52fdb8a7bdd61b20eae6333c26b0781cb53ea963.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.319ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} ^{+}}" loading="lazy"> ist bezüglich der Multiplikation eine kommutative Gruppe. Wegen der eindeutigen Primfaktorzerlegung lässt sich jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r\in \mathbb {Q} ^{+}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r\in \mathbb {Q} ^{+}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f609e38525092d8b949c7f68e901a339ee527b9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.208ex; height:2.843ex;" alt="{\displaystyle r\in \mathbb {Q} ^{+}}" loading="lazy"> eindeutig schreiben <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r=p_{1}^{z_{1}}\cdots p_{n}^{z_{n}}}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r=p_{1}^{z_{1}}\cdots p_{n}^{z_{n}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/958e897fd7e8edf0eb7aa86cbf5cd1ba2fea928d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.773ex; height:3.176ex;" alt="{\displaystyle r=p_{1}^{z_{1}}\cdots p_{n}^{z_{n}}}" loading="lazy"> mit Primzahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1},\dots ,p_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{1},\dots ,p_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7a90efd8373125c90affd49557696138859c165a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.879ex; height:2.009ex;" alt="{\displaystyle p_{1},\dots ,p_{n}}" loading="lazy">. Es ist also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} ^{+}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/52fdb8a7bdd61b20eae6333c26b0781cb53ea963.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.319ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} ^{+}}" loading="lazy"> eine freie abelsche Gruppe mit abzählbarer Basis.</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"> ist genau dann ein <a href="Schiefk%C3%B6rper" title="Schiefkörper">Schiefkörper</a>, wenn jeder Modul über diesem Ring frei ist.</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Der_Rang_eines_freien_Moduls">Der Rang eines freien Moduls</h3></div>

Viele der Sätze über Basen von Vektorräumen gelten bei freien Moduln nicht mehr:

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4d3a7c3cd696169e3bfa23c60c696ec0722ea2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle \textstyle V}" loading="lazy"> ein Vektorraum über dem Körper <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"> mit einer Basis von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f305e136d6d5ea97d6abc8d333894f20f297ed4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle \textstyle n}" loading="lazy"> Elementen, so ist jedes System von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f305e136d6d5ea97d6abc8d333894f20f297ed4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle \textstyle n}" loading="lazy"> freien Elementen auch ein Erzeugendensystem, also eine Basis. Über Ringen gilt das im Allgemeinen nicht: So ist beispielsweise im <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">-Modul <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"> die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle \{2\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \textstyle \{2\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd2e63d4db9dae9ed547b59c6da1d4330043743b.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 \textstyle \{2\}}" loading="lazy"> frei, aber keine Basis.</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> ein Vektorraum, so sind je zwei Basen gleich mächtig. Dies gilt noch bei kommutativen Ringen. Ist also der 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"> kommutativ und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{n}\cong R^{m}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8611f891537417d68d61f38dd3bb01a6e40e84f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.52ex; height:2.343ex;" alt="{\displaystyle R^{n}\cong R^{m}}" loading="lazy">, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=m}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/480d6131c6cb07a90f4ec18a376a59fab884b860.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.676ex;" alt="{\displaystyle n=m}" loading="lazy">. Einen kurzen relativ elementaren Beweis hierzu findet man in dem Buch von Jens Carsten Jantzen und <a href="Joachim_Schwermer" title="Joachim Schwermer">Joachim Schwermer</a>.<a href="#cite_note-2">[2]</a> Über nicht kommutativen 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"> ist der Satz im Allgemeinen falsch. Ein Beispiel ist die Menge der <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">-Endomorphismen eines freien <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">-Moduls mit unendlicher Basis. Man kann daher den Rang eines freien Moduls nicht allgemein definieren. Ringe, bei denen je zwei Basen eines freien Moduls gleich mächtig sind, heißen IBN-Ringe.<a href="#cite_note-3">[3]</a> <a href="Noetherscher_Ring" title="Noetherscher Ring">Noethersche Ringe</a> haben diese Eigenschaft.</li>

<li>Es gilt allgemeiner: Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \colon R\rightarrow S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23c8647373e7e90929d2354a331d9a6676ed5dcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.113ex; height:2.676ex;" alt="{\displaystyle \rho \colon R\rightarrow S}" loading="lazy"> ein Homomorphismus von Ringen und 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_/eb734a37dd21ce173a46342d1cc64c92/4611d85173cd3b508e67077d4a1252c9c05abca2.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"> ein IBN-Ring, so auch <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">. Gibt es also beispielsweise von <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"> einen Ringhomomorphismus nach einem noetherschen 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">, so ist <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"> ein IBN-Ring.</li></ol>

<div class="mw-heading mw-heading2"><h2 id="Eigenschaften_freier_Moduln">Eigenschaften freier Moduln</h2></div>

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

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (m_{i}|i\in I)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1fc4296e8684da276d85610137b2210ba0787823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.111ex; height:2.843ex;" alt="{\displaystyle (m_{i}|i\in I)}" loading="lazy"> eine Familie von Elementen aus 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">, so gibt es genau einen <a href="Modulhomomorphismus" title="Modulhomomorphismus">Homomorphismus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{(I)}=\bigoplus _{i\in I}Re_{i}\rightarrow M}">

<msup>

</mrow>

</msup>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R^{(I)}=\bigoplus _{i\in I}Re_{i}\rightarrow M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f309d551a2fdc1dd044c9c799d1d6c59f166fa4b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:20.804ex; height:5.676ex;" alt="{\displaystyle R^{(I)}=\bigoplus _{i\in I}Re_{i}\rightarrow M}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(e_{i})=m_{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/648baf68a79b7115de66835348c337e79829637e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.91ex; height:2.843ex;" alt="{\displaystyle f(e_{i})=m_{i}}" loading="lazy">. Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e_{i}|i\in I)}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (e_{i}|i\in I)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/edf6fcdb5a8149841c34258e1bfc51b72b6c53de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.154ex; height:2.843ex;" alt="{\displaystyle (e_{i}|i\in I)}" loading="lazy"> eine Basis (im Zweifel die kanonische) von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{(I)}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08c07dd98de2b46b8ccc59923f9012d98f255fcd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.104ex; height:2.843ex;" alt="{\displaystyle R^{(I)}}" loading="lazy">. Erzeugt die Familie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (m_{i}|i\in I)}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (m_{i}|i\in I)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1fc4296e8684da276d85610137b2210ba0787823.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.111ex; height:2.843ex;" alt="{\displaystyle (m_{i}|i\in I)}" loading="lazy"> den 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">, 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 Epimorphismus. Jeder Modul ist also epimorphes Bild eines freien Moduls.</li>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> ein freier Modul und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon M\rightarrow F}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/39ba4c9faa7b4f0a87df6451efd69866b21e5078.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.509ex;" alt="{\displaystyle f\colon M\rightarrow F}" loading="lazy"> ein Epimorphismus, so ist <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"> <a href="Untermodul" title="Untermodul">direkter Summand</a> in <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">. Es gibt ein <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon F\rightarrow M}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f651d51828de2c26b40a690876a3c4ff8a7a667.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.947ex; height:2.509ex;" alt="{\displaystyle g\colon F\rightarrow M}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ g=\mathbf {1} _{F}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b47195600061f5be328e3f8519dbeaa511bf3ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.488ex; height:2.509ex;" alt="{\displaystyle f\circ g=\mathbf {1} _{F}}" loading="lazy">.</li>

<li>Die Aussage 1. kann allgemeiner und zugleich genauer ausgedrückt werden. Zu jeder Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> gehört der freie Modul <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} (X):=R^{(X)}}">

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} (X):=R^{(X)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5bbb5e97a64163523803bfc6a23418e829f16330.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.893ex; height:3.343ex;" alt="{\displaystyle \mathbf {F} (X):=R^{(X)}}" loading="lazy"> und die kanonische injektive Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (X)\colon X\ni x\mapsto e_{x}\in R^{(X)}}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Phi (X)\colon X\ni x\mapsto e_{x}\in R^{(X)}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9560537b016f2ebbe1e35e1e930fba47eb3be40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.038ex; height:3.343ex;" alt="{\displaystyle \Phi (X)\colon X\ni x\mapsto e_{x}\in R^{(X)}}" loading="lazy">. Ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> eine weitere Menge und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \colon X\rightarrow Y}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/793cdf85bd5417072e25bd02aad3865139bd1669.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.889ex; height:2.176ex;" alt="{\displaystyle \alpha \colon X\rightarrow Y}" loading="lazy"> eine Abbildung zwischen den Mengen, so gibt es zu der Familie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e_{\alpha (x)}|x\in X)}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (e_{\alpha (x)}|x\in X)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bfcb20e98682f3fc6412c86022604e1652a3bc54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:13.194ex; height:3.176ex;" alt="{\displaystyle (e_{\alpha (x)}|x\in X)}" loading="lazy"> genau einen Homomorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} (\alpha )\colon \mathbf {F} (X)\rightarrow \mathbf {F} (Y)}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} (\alpha )\colon \mathbf {F} (X)\rightarrow \mathbf {F} (Y)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93af5634a963b792beccd37162d03526c05df8be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.365ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} (\alpha )\colon \mathbf {F} (X)\rightarrow \mathbf {F} (Y)}" loading="lazy">, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} (\alpha )\circ \Phi (X)=\Phi (Y)\circ \alpha }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} (\alpha )\circ \Phi (X)=\Phi (Y)\circ \alpha }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4aec61ba82cd238971665a368fe42abf39862cf9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.683ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} (\alpha )\circ \Phi (X)=\Phi (Y)\circ \alpha }" loading="lazy"> gilt. Das heißt, folgendes Diagramm ist kommutativ: Sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \colon X\rightarrow Y\,\beta \colon Y\rightarrow Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \alpha \colon X\rightarrow Y\,\beta \colon Y\rightarrow Z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2c4ef2e10ef085cb89e1432db379bbe73b503e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.71ex; height:2.509ex;" alt="{\displaystyle \alpha \colon X\rightarrow Y\,\beta \colon Y\rightarrow Z}" loading="lazy"> Abbildungen, so ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} (\alpha \circ \beta )=\mathbf {F} (\alpha )\circ \mathbf {F} (\beta )}">

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {F} (\alpha \circ \beta )=\mathbf {F} (\alpha )\circ \mathbf {F} (\beta )}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b9dae4635a09bd9c20c7105aa7a399b30fbf58e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.603ex; height:2.843ex;" alt="{\displaystyle \mathbf {F} (\alpha \circ \beta )=\mathbf {F} (\alpha )\circ \mathbf {F} (\beta )}" loading="lazy">. In der Sprache der <a href="Kategorientheorie" title="Kategorientheorie">Kategorientheorie</a> lässt sich das so ausdrücken: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da18bef8c979f3548bb0d8976f5844012d7b8256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} }" loading="lazy"> ist ein treuer Funktor von der Kategorie der Mengen in die Kategorie der freien Moduln. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aed80a2011a3912b028ba32a52dfa57165455f24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }" loading="lazy"> ist ein funktorieller Monomorphismus zwischen dem Identitätsfunktor und dem Funktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da18bef8c979f3548bb0d8976f5844012d7b8256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} }" loading="lazy">.</li>

<li>Wie in 3. gehört zu jedem 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"> der freie Modul <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(M)=R^{(M)}=\bigoplus _{m\in M}Re_{m}}">

<msup>

</mrow>

</msup>

</mrow>

</munder>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(M)=R^{(M)}=\bigoplus _{m\in M}Re_{m}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/42d326e0168a2820904d29d5985dc755e8186c0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:26.367ex; height:5.676ex;" alt="{\displaystyle F(M)=R^{(M)}=\bigoplus _{m\in M}Re_{m}}" loading="lazy">. Dazu gehört der eindeutig bestimmte Epimorphismus <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (M):F(M)\ni e_{m}\mapsto m\in M}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi (M):F(M)\ni e_{m}\mapsto m\in M}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5d6d2eb35af1a51fd2ab5f9dd38a6b2f97b51f5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.526ex; height:2.843ex;" alt="{\displaystyle \Psi (M):F(M)\ni e_{m}\mapsto m\in M}" loading="lazy">. Für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \colon M\rightarrow N}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a1f2ab8531b78f71cb1b02aea42b57f54440efdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.641ex; height:2.176ex;" alt="{\displaystyle \alpha \colon M\rightarrow N}" loading="lazy"> ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi (N)\circ F(\alpha )=\alpha \circ \Psi (N)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Psi (N)\circ F(\alpha )=\alpha \circ \Psi (N)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9f3a2df02b356595f117799517dafd5e38f676c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.375ex; height:2.843ex;" alt="{\displaystyle \Psi (N)\circ F(\alpha )=\alpha \circ \Psi (N)}" loading="lazy">. Es ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Psi }">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5471531a3fe80741a839bc98d49fae862a6439a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Psi }" loading="lazy"> ein funktorieller Epimorphismus zwischen dem Funktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {F} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/da18bef8c979f3548bb0d8976f5844012d7b8256.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.683ex; height:2.176ex;" alt="{\displaystyle \mathbf {F} }" loading="lazy"> und dem Identitätsfunktor.</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Freie_Moduln_über_besonderen_Ringen">Freie Moduln über besonderen Ringen</h3></div>

<ol><li>Über <a href="Hauptidealring" title="Hauptidealring">Hauptidealringen</a> ist jeder Untermodul eines freien Moduls wieder frei.</li>

<li>Über <a href="Lokaler_Ring" title="Lokaler Ring">lokalen Ringen</a> sind alle <a href="Untermodul" title="Untermodul">direkte Summanden </a> von freien Moduln (das sind <a href="Projektives_Objekt" title="Projektives Objekt">projektive Moduln</a>) frei.</li></ol>

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

Zu jeder Menge <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 jedem 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"> gibt es den freien <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">-Linksmodul <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FS}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a95114882d7c570a8be6b2ca10917026396337e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.24ex; height:2.176ex;" alt="{\displaystyle FS}" loading="lazy"> 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">. Sein Träger ist die Menge der formalen Linearkombinationen von <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">-Elementen, kodiert etwa als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FS:=\{v\colon S\to R\mid \{s\in S\mid v(s)\neq 0\}{\text{ endlich}}\}}">

<mtext> endlich</mtext>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle FS:=\{v\colon S\to R\mid \{s\in S\mid v(s)\neq 0\}{\text{ endlich}}\}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/61e1be2eb3bcf620f1aefc8bc9434bb6716d4531.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.084ex; height:2.843ex;" alt="{\displaystyle FS:=\{v\colon S\to R\mid \{s\in S\mid v(s)\neq 0\}{\text{ endlich}}\}}" loading="lazy">.

Addition und Skalarmultiplikation erfolgen dabei punktweise:

<table>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe6ef363cd19902d1a7a71fb1c8b21e8ede52406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle +}" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2e6c5693c4d5334c475a3c5c696d20f65ec7993.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle \colon }" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/06596635950246c681c2a2e441c21a93b244c19d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.175ex; height:2.176ex;" alt="{\displaystyle FS\times FS\to FS}" loading="lazy"></td>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (v+w)(s):=v(s)+w(s)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3a88882a87d8cfb6329f7b9068558a574cb3949.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.518ex; height:2.843ex;" alt="{\displaystyle (v+w)(s):=v(s)+w(s)}" loading="lazy">

</td></tr>

<tr>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba2c023bad1bd39ed49080f729cbf26bc448c9ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="{\displaystyle \cdot }" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2e6c5693c4d5334c475a3c5c696d20f65ec7993.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle \colon }" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8de0bca20074dbdbc93f0e0a9a7b0b0d15c958aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.699ex; height:2.176ex;" alt="{\displaystyle R\times FS\to FS}" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c068391152c82fc934b2be57772640c6e984da15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.065ex; height:2.843ex;" alt="{\displaystyle (r\cdot v)(s):=r\cdot v(s)}" loading="lazy">

</td></tr></tbody></table>

Die Elemente von <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"> sind hierbei keine Elemente von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle FS}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a95114882d7c570a8be6b2ca10917026396337e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.24ex; height:2.176ex;" alt="{\displaystyle FS}" loading="lazy">. Wenn <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"> eine <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1=:e}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6fe0c087a1d8e211c39cbd7502948d871770395e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.991ex; height:2.176ex;" alt="{\displaystyle 1=:e}" loading="lazy"> (oder auch nur eine Links-Erzeugende <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} e+Re=R}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/27f430a2fd3c241aa4f3f2c1a31a2b261fffd56a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.184ex; height:2.343ex;" alt="{\displaystyle \mathbb {Z} e+Re=R}" loading="lazy">) hat, so lassen sie sich aber einbetten mittels

<table>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/898c8e8333aef475cfec45def820e51b41120e15.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.448ex; height:2.176ex;" alt="{\displaystyle \eta _{S}}" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f2e6c5693c4d5334c475a3c5c696d20f65ec7993.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:1.676ex;" alt="{\displaystyle \colon }" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff50ca587e26249c0ad1ed1037a121e796254d85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.353ex; height:2.176ex;" alt="{\displaystyle S\to FS}" loading="lazy"></td>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \eta _{S}(s):}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b4b65c382b65633700c1dd57ed49aa7fa3f5151.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.64ex; height:2.843ex;" alt="{\displaystyle \eta _{S}(s):}" loading="lazy"></td>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/967820610bab9444b66866ca77023f494a6e61e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.877ex; height:2.176ex;" alt="{\displaystyle S\to R}" loading="lazy">

</td></tr>

<tr>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mtext>sonst.</mtext>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t\mapsto {\begin{cases}e&s=t\\0&{\text{sonst.}}\end{cases}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf423a6fe43d6065d0668d20297db047fe1035cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.272ex; height:6.176ex;" alt="{\displaystyle t\mapsto {\begin{cases}e&s=t\\0&{\text{sonst.}}\end{cases}}}" loading="lazy">

</td></tr></tbody></table>

Der freie <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">-Rechtsmodul ist der freie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{\text{op}}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d46af7f0301d04fef72be7801bc7edf4173963ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.732ex; height:2.343ex;" alt="{\displaystyle R^{\text{op}}}" loading="lazy">-Linksmodul, wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{\text{op}}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d46af7f0301d04fef72be7801bc7edf4173963ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.732ex; height:2.343ex;" alt="{\displaystyle R^{\text{op}}}" loading="lazy"> den <a href="Gegenring" title="Gegenring">Gegenring</a> von <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"> bezeichnet.

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

Das folgende Diagramm setzt die Freiheit eines Moduls <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"> über einem kommutativen Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> mit den Eigenschaften <a href="Projektiver_Modul" class="mw-redirect" title="Projektiver Modul">projektiv</a>, <a href="Flachheit_(Algebra)" title="Flachheit (Algebra)">flach</a> und <a href="Torsion_(Algebra)" title="Torsion (Algebra)">torsionsfrei</a> in Beziehung:

<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>

<ul><li><a href="Basis_(Vektorraum)" title="Basis (Vektorraum)">Basis (Vektorraum)</a></li>

<li><a href="Projektives_Objekt" title="Projektives Objekt">Projektives Objekt</a></li></ul>

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

<ul><li>Tsit-Yuen Lam: Lectures on modules and rings. GTM 189, Springer, 1999, ISBN 0-387-98428-3.</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 class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Tsit-Yuen Lam: Lectures on modules and rings. GTM 189, Springer, 1999, ISBN 0-387-98428-3, S. 22 f.

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Jens Carsten Jantzen, Joachim Schwermer: Algebra, Springer, 2006, ISBN 3-540-21380-5, <a href="https://doi.org/10.1007/3-540-29287-X" class="extiw external" title="doi:10.1007/3-540-29287-X">doi:10.1007/3-540-29287-X</a>, Seite 165

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Siehe hierzu den Artikel <a href="https://en.wikipedia.org/wiki/Invariant_basis_number" class="extiw external" title="en:Invariant basis number">en:Invariant basis number</a>

</li>

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