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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Module quotient</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="fr" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><p>En <a href="Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, un <b>module quotient</b> est le <a href="Module_sur_un_anneau" title="Module sur un anneau">module</a> obtenu en <a href="Relation_d'%C3%A9quivalence" title="Relation d'équivalence">quotientant</a> un module sur un <a href="Anneau_unitaire" title="Anneau unitaire">anneau</a> par un de ses <a href="Sous-module" class="mw-redirect" title="Sous-module">sous-modules</a>.
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<div class="mw-heading mw-heading2"><h2 id="Définition"><span id="D.C3.A9finition"></span>Définition</h2></div>
<p>Soient <i>M</i> un <a href="Module_sur_un_anneau" title="Module sur un anneau">module</a> sur un <a href="Anneau_unitaire" title="Anneau unitaire">anneau</a> <i>A</i> et <i>N</i> un <a href="Sous-module" class="mw-redirect" title="Sous-module">sous-module</a> de <i>M</i>.
</p><p>Le <a href="Groupe_(math%C3%A9matiques)" title="Groupe (mathématiques)">groupe</a> (<i>M</i>,+) étant <a href="Groupe_ab%C3%A9lien" title="Groupe abélien">abélien</a>, son <a href="Sous-groupe" title="Sous-groupe">sous-groupe</a> (<i>N</i>,+) est <a href="Sous-groupe_normal" title="Sous-groupe normal">normal</a>, ce qui permet de définir le <a href="Groupe_quotient" title="Groupe quotient">groupe quotient</a> (<i>M/N</i>,+).
</p><p>Sur ce groupe (<i>M/N</i>,+), qui est abélien, il existe une unique <a href="Loi_externe" class="mw-redirect" title="Loi externe">loi externe</a> faisant de <i>M/N</i> un <i>A</i>-module et telle que la projection canonique <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :M\rightarrow M/N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>π<!-- π --></mi>
<mo>:</mo>
<mi>M</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
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<mi>N</mi>
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<annotation encoding="application/x-tex">{\displaystyle \pi :M\rightarrow M/N}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d0ee06c508ba589538a146f97ae6061328dd66.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.994ex; height:2.843ex;" alt="{\displaystyle \pi :M\rightarrow M/N}" loading="lazy"></span> soit non seulement un <a href="Morphisme_de_groupes" title="Morphisme de groupes">morphisme de groupes</a>, mais un <a href="Morphisme" title="Morphisme">morphisme</a> de <i>A</i>-modules :
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \forall a\in A,~\forall m\in M,\qquad a.(m+N)=(am)+N~.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mi>a</mi>
<mo>∈<!-- ∈ --></mo>
<mi>A</mi>
<mo>,</mo>
<mtext> </mtext>
<mi mathvariant="normal">∀<!-- ∀ --></mi>
<mi>m</mi>
<mo>∈<!-- ∈ --></mo>
<mi>M</mi>
<mo>,</mo>
<mspace width="2em"></mspace>
<mi>a</mi>
<mo>.</mo>
<mo stretchy="false">(</mo>
<mi>m</mi>
<mo>+</mo>
<mi>N</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mi>m</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>N</mi>
<mtext> </mtext>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \forall a\in A,~\forall m\in M,\qquad a.(m+N)=(am)+N~.}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be1562cad46c6290dbf35bd572a289d0bf0e10f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.342ex; height:2.843ex;" alt="{\displaystyle \forall a\in A,~\forall m\in M,\qquad a.(m+N)=(am)+N~.}" loading="lazy"></span></dd></dl>
<div class="mw-heading mw-heading2"><h2 id="Exemples">Exemples</h2></div>
<ul><li><i>M/M</i> est le module trivial {0}.</li>
<li><i>M</i>/{0} est <a href="Isomorphe" class="mw-redirect" title="Isomorphe">isomorphe</a> à <i>M</i>.</li>
<li>Si <i>M</i> est égal à l'anneau <i>A</i> (vu comme module à gauche sur lui-même), ses sous-modules sont les <a href="Id%C3%A9al_%C3%A0_gauche" class="mw-redirect" title="Idéal à gauche">idéaux à gauche</a> de <i>A</i>. Le module quotient de <i>A</i> par un idéal bilatère <i>I</i> est l'<a href="Anneau_quotient" title="Anneau quotient">anneau quotient</a> <i>A/I</i>, vu comme <i>A</i>-module.</li>
<li>Si <i>I</i> est un idéal bilatère de <i>A</i>, la structure de <i>A</i>-module du quotient de <i>M</i> par le sous-module</li></ul>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle IM=\{\sum _{j=1}^{n}a_{j}m_{j}~|~n\in \mathbb {N} ,~a_{1},\ldots ,a_{n}\in I,~m_{1},\ldots ,m_{n}\in M\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
<mi>M</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<munderover>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
<mo>=</mo>
<mn>1</mn>
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<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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</munderover>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
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</msub>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
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</msub>
<mtext> </mtext>
<mrow class="MJX-TeXAtom-ORD">
<mo stretchy="false">|</mo>
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<mtext> </mtext>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">N</mi>
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<mo>,</mo>
<mtext> </mtext>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
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<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>a</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>∈<!-- ∈ --></mo>
<mi>I</mi>
<mo>,</mo>
<mtext> </mtext>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
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</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo>,</mo>
<msub>
<mi>m</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
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<mo>∈<!-- ∈ --></mo>
<mi>M</mi>
<mo fence="false" stretchy="false">}</mo>
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<annotation encoding="application/x-tex">{\displaystyle IM=\{\sum _{j=1}^{n}a_{j}m_{j}~|~n\in \mathbb {N} ,~a_{1},\ldots ,a_{n}\in I,~m_{1},\ldots ,m_{n}\in M\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f01388ea1b230106ec93537ee194c0300237947.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:59.557ex; height:7.176ex;" alt="{\displaystyle IM=\{\sum _{j=1}^{n}a_{j}m_{j}~|~n\in \mathbb {N} ,~a_{1},\ldots ,a_{n}\in I,~m_{1},\ldots ,m_{n}\in M\}}" loading="lazy"></span></dd></dl>
<p>est induite par sa structure naturelle de <i>A/I</i>-module.
</p>
<div class="mw-heading mw-heading2"><h2 id="Propriétés"><span id="Propri.C3.A9t.C3.A9s"></span>Propriétés</h2></div>
<p>Tout morphisme de <i>A</i>-modules <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:M\rightarrow L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>:</mo>
<mi>M</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f:M\rightarrow L}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cb4349191cf52061d8a9e05305bfea9e655b658f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.855ex; height:2.509ex;" alt="{\displaystyle f:M\rightarrow L}" loading="lazy"></span> dont le <a href="Noyau_(alg%C3%A8bre)" title="Noyau (algèbre)">noyau</a> contient <i>N</i> se <a href="Th%C3%A9or%C3%A8me_de_factorisation" title="Théorème de factorisation">factorise</a> de façon unique par <i>M/N</i>, c'est-à-dire qu'il existe un unique morphisme de <i>A</i>-modules <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {f}}:M/N\to L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>f</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo>:</mo>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>N</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}:M/N\to L}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e568659b7a684649463b7e92e1bd676ad9ba403b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.501ex; height:3.176ex;" alt="{\displaystyle {\tilde {f}}:M/N\to L}" loading="lazy"></span> tel que <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tilde {f}}\circ \pi =f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mover>
<mi>f</mi>
<mo stretchy="false">~<!-- ~ --></mo>
</mover>
</mrow>
</mrow>
<mo>∘<!-- ∘ --></mo>
<mi>π<!-- π --></mi>
<mo>=</mo>
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\tilde {f}}\circ \pi =f}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ad36667485b15dfcd85b05974f0354ac22d68b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.603ex; height:3.009ex;" alt="{\displaystyle {\tilde {f}}\circ \pi =f}" loading="lazy"></span>.
</p>
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