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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Noetherscher Ring</span></h1>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>In der <a href="Algebra" title="Algebra">Algebra</a> werden bestimmte Strukturen (<a href="Ring_(Algebra)" title="Ring (Algebra)">Ringe</a> und <a href="Modul_(Mathematik)" title="Modul (Mathematik)">Moduln</a>) <b>noethersch</b> genannt, wenn sie <i>keine</i> unendliche Schachtelung von immer größeren Unterstrukturen enthalten können. Der Begriff ist nach der Mathematikerin <a href="Emmy_Noether" title="Emmy Noether">Emmy Noether</a> benannt.
</p>
<div class="mw-heading mw-heading2"><h2 id="Noethersche_Moduln">Noethersche Moduln</h2></div>
<p>Es sei <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> ein <a href="Unit%C3%A4rer_Ring" class="mw-redirect" title="Unitärer Ring">unitärer Ring</a> (d. h. ein Ring mit Einselement). Ein <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>-<a href="Linksmodul" class="mw-redirect" title="Linksmodul">Linksmodul</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> heißt <i>noethersch</i>, wenn er eine der folgenden äquivalenten Bedingungen erfüllt:
</p>
<ul><li>Jeder <a href="Untermodul" title="Untermodul">Untermodul</a> ist endlich erzeugt.</li>
<li>(Aufsteigende Kettenbedingung) Jede unendliche aufsteigende Kette</li></ul>
<dl><dd><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 N_{1}\subseteq N_{2}\subseteq N_{3}\subseteq \dotsb }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>N</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>⊆<!-- ⊆ --></mo>
<msub>
<mi>N</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>⊆<!-- ⊆ --></mo>
<msub>
<mi>N</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo>⊆<!-- ⊆ --></mo>
<mo>⋯<!-- ⋯ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N_{1}\subseteq N_{2}\subseteq N_{3}\subseteq \dotsb }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7979d3655a1d96493ddb5065d670170ae3de5224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.78ex; height:2.509ex;" alt="{\displaystyle N_{1}\subseteq N_{2}\subseteq N_{3}\subseteq \dotsb }" loading="lazy"></span></dd></dl></dd>
<dd>von Untermoduln wird stationär, d. h., es gibt einen Index <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n}</annotation>
</semantics>
</math></span><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"></span>, so dass
<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 N_{n}=N_{n+1}=N_{n+2}=\dotsb }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>N</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>N</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
<mo>=</mo>
<msub>
<mi>N</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msub>
<mo>=</mo>
<mo>⋯<!-- ⋯ --></mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle N_{n}=N_{n+1}=N_{n+2}=\dotsb }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/681c39d186f678b0b3d7b747aeebd5b1372ea5e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.474ex; height:2.509ex;" alt="{\displaystyle N_{n}=N_{n+1}=N_{n+2}=\dotsb }" loading="lazy"></span></dd></dl></dd></dl>
<ul><li>(Maximalbedingung für Untermoduln) Jede nichtleere Menge von Untermoduln von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> hat ein <a href="Maximales_Element" class="mw-redirect" title="Maximales Element">maximales Element</a> bezüglich Inklusion.</li></ul>
<div class="mw-heading mw-heading3"><h3 id="Beispiele">Beispiele</h3></div>
<ul><li>Jeder endliche Modul ist noethersch.</li>
<li>Jeder endlich erzeugte Modul über einem noetherschen Ring ist noethersch.</li>
<li>Jede endliche direkte Summe noetherscher Moduln ist noethersch.</li>
<li><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 \mathbb {Z} \left[{\tfrac {1}{p}}\right]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mrow>
<mo>[</mo>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mn>1</mn>
<mi>p</mi>
</mfrac>
</mstyle>
</mrow>
<mo>]</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \left[{\tfrac {1}{p}}\right]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b619d36f1fa360fbf4958f34dd6e18544bdfb0f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.795ex; height:4.843ex;" alt="{\displaystyle \mathbb {Z} \left[{\tfrac {1}{p}}\right]}" loading="lazy"></span> ist <b>nicht</b> noethersch als <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 \mathbb {Z} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>
</semantics>
</math></span><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"></span>-Modul.</li></ul>
<div class="mw-heading mw-heading3"><h3 id="Eigenschaften">Eigenschaften</h3></div>
<ul><li>Jeder <a href="Surjektivit%C3%A4t" class="mw-redirect" title="Surjektivität">surjektive</a> <a href="Endomorphismus" title="Endomorphismus">Endomorphismus</a> ist ein <a href="Automorphismus" title="Automorphismus">Automorphismus</a>.</li>
<li>Für eine <a href="Kurze_exakte_Sequenz" class="mw-redirect" title="Kurze exakte Sequenz">kurze exakte Sequenz</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \quad 0\rightarrow M_{1}\rightarrow M_{2}\rightarrow M_{3}\rightarrow 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mspace width="1em"></mspace>
<mn>0</mn>
<mo stretchy="false">→<!-- → --></mo>
<msub>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo stretchy="false">→<!-- → --></mo>
<msub>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo stretchy="false">→<!-- → --></mo>
<msub>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
<mo stretchy="false">→<!-- → --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \quad 0\rightarrow M_{1}\rightarrow M_{2}\rightarrow M_{3}\rightarrow 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e54c29dee32ec5fb085a4b4c5aed67f1114a43c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.029ex; height:2.509ex;" alt="{\displaystyle \quad 0\rightarrow M_{1}\rightarrow M_{2}\rightarrow M_{3}\rightarrow 0}" loading="lazy"></span> sind äquivalent:
<ol><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{2}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M_{2}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0d5d4dffae5ee0db4cc433e252ee9ed7530e5cf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.308ex; height:2.509ex;" alt="{\displaystyle M_{2}}" loading="lazy"></span> ist noethersch.</li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M_{1},M_{3}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>M</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>3</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M_{1},M_{3}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87afb0d19570f8d817e75846238df5afe4e77111.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.651ex; height:2.509ex;" alt="{\displaystyle M_{1},M_{3}}" loading="lazy"></span> sind noethersch.</li></ol></li>
<li>Ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><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"></span> ein <a href="Vektorraum" title="Vektorraum">Vektorraum</a>, so ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><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"></span> genau dann noethersch, wenn er endlich-dimensional ist. In diesem Fall ist der Modul auch <a href="Artinsch" class="mw-redirect" title="Artinsch">artinsch</a>.</li>
<li>Ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> linksnoethersch, das <a href="Jacobson-Radikal" title="Jacobson-Radikal">Jacobson-Radikal</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J=\operatorname {Rad} (R)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>J</mi>
<mo>=</mo>
<mi>Rad</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>R</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle J=\operatorname {Rad} (R)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd1b1c4eccced38a7c5b61d224f88d2c93bb70f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.309ex; height:2.843ex;" alt="{\displaystyle J=\operatorname {Rad} (R)}" loading="lazy"></span> <a href="Nilpotenz" class="mw-redirect" title="Nilpotenz">nilpotent</a> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R/J}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>J</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R/J}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5c7fa454e6abcf300c7e857627bf56ea72f0f1fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.398ex; height:2.843ex;" alt="{\displaystyle R/J}" loading="lazy"></span> halbeinfach, dann ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> auch <a href="Artinsch" class="mw-redirect" title="Artinsch">linksartinsch</a>.</li>
<li>Über einem noetherschen Ring ist jeder endlich erzeugte Modul auch <a href="Endliche_Pr%C3%A4sentierbarkeit_(Modul)" title="Endliche Präsentierbarkeit (Modul)">endlich präsentiert</a> (die Umkehrung gilt immer).</li>
<li>Die endlich erzeugten Moduln über einem noetherschen Ring bilden eine <a href="Abelsche_Kategorie" title="Abelsche Kategorie">abelsche Kategorie</a>; die Voraussetzung, dass der Ring noethersch ist, ist dabei essentiell.</li>
<li>Jeder echte Untermodul eines noetherschen Moduls besitzt eine <a href="Prim%C3%A4rzerlegung" title="Primärzerlegung">Primärzerlegung</a>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Noethersche_Ringe">Noethersche Ringe</h2></div>
<p>Ein Ring <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> heißt
</p>
<ul><li><i>linksnoethersch</i>, wenn er als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>-Linksmodul noethersch ist;</li>
<li><i>rechtsnoethersch</i>, wenn er als <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span>-Rechtsmodul noethersch ist;</li>
<li><i>noethersch</i>, wenn er links- und rechtsnoethersch ist.</li></ul>
<p>Bei <a href="Kommutativ" class="mw-redirect" title="Kommutativ">kommutativen</a> Ringen sind alle drei Begriffe identisch und äquivalent dazu, dass alle <a href="Ideal_(Ringtheorie)" title="Ideal (Ringtheorie)">Ideale</a> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> endlich erzeugt sind.
</p>
<div class="mw-heading mw-heading3"><h3 id="Beispiele_2">Beispiele</h3></div>
<ul><li><a href="Artinscher_Ring" class="mw-redirect" title="Artinscher Ring">Artinsche Ringe</a> sind noethersch.</li>
<li><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 \mathbb {Z} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation>
</semantics>
</math></span><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"></span> ist noethersch, aber nicht artinsch.</li>
<li><a href="Faktorring" title="Faktorring">Quotienten</a> und <a href="Lokalisierung_(Algebra)" title="Lokalisierung (Algebra)">Lokalisierungen</a> noetherscher Ringe sind noethersch.</li>
<li><a href="Hauptidealring" title="Hauptidealring">Hauptidealringe</a> oder allgemeiner <a href="Dedekindring" title="Dedekindring">Dedekindringe</a> sind noethersch.</li>
<li>Ist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> ein noetherscher Ring, so ist auch der Polynomring <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R[X]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>R</mi>
<mo stretchy="false">[</mo>
<mi>X</mi>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle R[X]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afeccd52deabb878398a8485755c3ceea80caf9a.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 R[X]}" loading="lazy"></span> noethersch (<a href="Hilbertscher_Basissatz" title="Hilbertscher Basissatz">Hilbertscher Basissatz</a>).</li>
<li>Daraus folgt, dass allgemein endlich erzeugte Algebren über einem noetherschen Ring wieder noethersch sind. Insbesondere sind endlich erzeugte Algebren über Körpern noethersch.</li>
<li>Der Polynomring <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 \mathbb {C} [X_{1},X_{2},\ldots ]}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo stretchy="false">[</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
</mrow>
</msub>
<mo>,</mo>
<msub>
<mi>X</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msub>
<mo>,</mo>
<mo>…<!-- … --></mo>
<mo stretchy="false">]</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} [X_{1},X_{2},\ldots ]}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fc62cd9ba6be2906e109f0167aacfd3fa3d4bf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.72ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} [X_{1},X_{2},\ldots ]}" loading="lazy"></span> in unendlich vielen Unbestimmten ist nicht noethersch, da das Ideal, das von allen Unbestimmten erzeugt wird, nicht endlich erzeugt ist.</li>
<li>Der <a href="Matrizenring" title="Matrizenring">Matrizenring</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{pmatrix}\mathbb {Z} &\mathbb {Q} \\0&\mathbb {Q} \end{pmatrix}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mo>(</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Z</mi>
</mrow>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Q</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">Q</mi>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}\mathbb {Z} &\mathbb {Q} \\0&\mathbb {Q} \end{pmatrix}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b78000bc4c73b9929cdb6abe526d42a1296f3c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.854ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}\mathbb {Z} &\mathbb {Q} \\0&\mathbb {Q} \end{pmatrix}}}" loading="lazy"></span> ist rechtsnoethersch, aber weder linksartinsch noch linksnoethersch.</li></ul>
<div class="mw-heading mw-heading3"><h3 id="Eigenschaften_2">Eigenschaften</h3></div>
<ul><li>Jedes <a href="Irreduzibles_Ideal" title="Irreduzibles Ideal">irreduzible Ideal</a> in einem noetherschen Ring ist ein <a href="Prim%C3%A4res_Ideal" title="Primäres Ideal">primäres Ideal</a>.</li>
<li>In einem noetherschen Ring kann jedes echte Ideal als Schnitt endlich vieler irreduzibler Ideale dargestellt werden. Insbesondere existiert in noetherschen Ringen eine <a href="Prim%C3%A4rzerlegung" title="Primärzerlegung">Primärzerlegung</a>.</li>
<li>In einem noetherschen Ring gibt es nur endlich viele minimale Primideale.</li>
<li>Jede von Null verschiedene Nichteinheit in einem noetherschen Ring lässt sich als endliches Produkt irreduzibler Elemente schreiben. Insbesondere ist ein noetherscher Ring, in welchem alle irreduziblen Element Primelemente sind, ein <a href="Faktorieller_Ring" title="Faktorieller Ring">faktorieller Ring</a>.</li>
<li>Ist in einem Ring das <a href="Nullideal" class="mw-redirect" title="Nullideal">Nullideal</a> Produkt <a href="Maximales_Ideal" title="Maximales Ideal">maximaler Ideale</a>, so ist der Ring genau dann noethersch, wenn er <a href="Artinsch" class="mw-redirect" title="Artinsch">artinsch</a> ist.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Noetherscher_Raum" title="Noetherscher Raum">Noetherscher Raum</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Emmy_Noether" title="Emmy Noether">Emmy Noether</a>: <i>Idealtheorie in Ringbereichen.</i> In: <i><a href="Mathematische_Annalen" title="Mathematische Annalen">Mathematische Annalen</a>.</i> 83, 1921, S. 24–66, <a rel="nofollow" class="external text" href="https://gdz.sub.uni-goettingen.de/id/PPN235181684_0083">GDZ</a></li>
<li><a href="Nicolas_Bourbaki" title="Nicolas Bourbaki">Nicolas Bourbaki</a>: <i>Algèbre commutative.</i> Band 8/9: <i>Chapitre 8: Dimension. Chapitre 9: Anneaux locaux noethériens complets.</i> Masson, Paris 1983, ISBN 2-225-78716-6 (<i>Éléments de mathématique</i>).</li>
<li>David Eisenbud: <i>Commutative Algebra with a View Toward Algebraic Geometry.</i> Corrected 3rd printing. Springer-Verlag, New York NY 1999, ISBN 0-387-94268-8 (<i>Graduate Texts in Mathematics</i> 150), (engl.).</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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