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<title>Primideal</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Primideal</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">In der <a href="Ringtheorie" class="mw-redirect" title="Ringtheorie">Ringtheorie</a> ist ein Primideal eine <a href="Teilmenge" title="Teilmenge">Teilmenge</a> eines <a href="Ringtheorie" class="mw-redirect" title="Ringtheorie">Ringes</a>, die sich ähnlich wie eine <a href="Primzahl" title="Primzahl">Primzahl</a> als Element der ganzen Zahlen verhält.

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

Es sei <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"> ein <a href="Ringtheorie" class="mw-redirect" title="Ringtheorie">Ring</a>. Dann heißt ein <a href="Ideal_(Ringtheorie)" title="Ideal (Ringtheorie)">zweiseitiges Ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}\subseteq R}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a6aa31fc3a10571725628ffa9cd35abdd83d7ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.025ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {p}}\subseteq R}" loading="lazy"> Primideal oder prim, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> <a href="Ideal_(Ringtheorie)#Besondere_Ideale" title="Ideal (Ringtheorie)">echt</a> ist, also <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}\neq R}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b9d534c4c2e5c47620d93155c8045b8588a44f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {p}}\neq R}" loading="lazy">, und wenn für alle Ideale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a,b}}\subseteq R}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66298600c3edeac13da5430e366866148ba12227.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.252ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {a,b}}\subseteq R}" loading="lazy"> gilt:<a href="#cite_note-1">[1]</a>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d520bd06db40eac4fa12e9404b1090a8607faa59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.616ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {ab}}\subseteq {\mathfrak {p}}}" loading="lazy"> folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}\subseteq {\mathfrak {p}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/70b514f94d0d1a0f264a61b9385fba8e1a84323f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.423ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {a}}\subseteq {\mathfrak {p}}}" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {b}}\subseteq {\mathfrak {p}}.}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a79254d71f6f1cf1ebbde39aca949f6d377401e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.1ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {b}}\subseteq {\mathfrak {p}}.}" loading="lazy"></dd></dl>

Außerdem heißt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> vollständiges Primideal oder vollprim, falls <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> echt ist und wenn für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in R}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d992800adce8f580e87a3c79b62f0e12d5349e1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.866ex; height:2.509ex;" alt="{\displaystyle a,b\in R}" loading="lazy"> gilt:

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6ff01b9eedcd23e64f0ed5b6de9480fc318c7fc2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.23ex; height:2.509ex;" alt="{\displaystyle ab\in {\mathfrak {p}}}" loading="lazy"> folgt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in {\mathfrak {p}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb493e9fec848c305ad2638b6ecff7e27c6f5185.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.233ex; height:2.176ex;" alt="{\displaystyle a\in {\mathfrak {p}}}" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in {\mathfrak {p}}.}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9036ba2ab0e3decbc2678ebb53f82cdf8811dcd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.648ex; height:2.509ex;" alt="{\displaystyle b\in {\mathfrak {p}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Äquivalente_Definitionen">Äquivalente Definitionen</h3></div>

<ul><li>Ein zweiseitiges Ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}\subseteq R}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a6aa31fc3a10571725628ffa9cd35abdd83d7ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.025ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {p}}\subseteq R}" loading="lazy"> ist genau dann prim, falls es echt ist und wenn für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a,b\in R}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d992800adce8f580e87a3c79b62f0e12d5349e1d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.866ex; height:2.509ex;" alt="{\displaystyle a,b\in R}" loading="lazy"> gilt:</li></ul>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca49c66b5e9b5f32249a737e4429c3df136c33f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.653ex; height:2.176ex;" alt="{\displaystyle r\in R}" loading="lazy"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle arb\in {\mathfrak {p}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/776fefbef91cb81f961b5fc9f75d206874602459.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.279ex; height:2.509ex;" alt="{\displaystyle arb\in {\mathfrak {p}}}" loading="lazy">) folgt (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a\in {\mathfrak {p}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb493e9fec848c305ad2638b6ecff7e27c6f5185.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.233ex; height:2.176ex;" alt="{\displaystyle a\in {\mathfrak {p}}}" loading="lazy"> oder <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b\in {\mathfrak {p}}}">

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</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3f0deea2d03b54faf3e3300eae370dfe03234466.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.001ex; height:2.509ex;" alt="{\displaystyle b\in {\mathfrak {p}}}" loading="lazy">).</dd></dl>

<ul><li>Ein zweiseitiges Ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}\subseteq R}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a6aa31fc3a10571725628ffa9cd35abdd83d7ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.025ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {p}}\subseteq R}" loading="lazy"> ist genau dann vollprim, falls es echt ist und wenn der <a href="Faktorring" title="Faktorring">Faktorring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R/{\mathfrak {p}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9da6e6f1e63ba7c169bec572ae2ca16f1c58936.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.089ex; height:2.843ex;" alt="{\displaystyle R/{\mathfrak {p}}}" loading="lazy"> <a href="Nullteiler" title="Nullteiler">nullteilerfrei</a> ist.</li></ul>

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

Die Menge aller (echten) Primideale eines Rings <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 <a href="Spektrum_eines_Ringes" title="Spektrum eines Ringes">Spektrum</a> von <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"> und wird mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Spec} (R)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b3eb3e8baab788aae477cd5fb3f4adde00599134.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.223ex; height:2.843ex;" alt="{\displaystyle \mathrm {Spec} (R)}" loading="lazy"> notiert.

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

<ul><li>Jedes vollprime Ideal ist prim, aber nicht umgekehrt. Zum Beispiel ist das <a href="Nullideal" class="mw-redirect" title="Nullideal">Nullideal</a> im <a href="Matrizenring" title="Matrizenring">Ring der reellen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f8a0e3400ffb97d67c00267ed50cddfe824cbe80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}" loading="lazy">-Matrizen</a> prim, aber nicht vollprim.</li>

<li>In kommutativen Ringen sind prim und vollprim äquivalent.</li></ul>

In kommutativen Ringen <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 gilt:

<ul><li>Ein Element <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\in R\backslash \left\{0\right\}}">

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<annotation encoding="application/x-tex">{\displaystyle p\in R\backslash \left\{0\right\}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c63cdcf4157a6fae5492dfb226d79030860562db.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:10.9ex; height:2.843ex;" alt="{\displaystyle p\in R\backslash \left\{0\right\}}" loading="lazy"> ist genau dann ein <a href="Primelement" title="Primelement">Primelement</a>, wenn das von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> erzeugte <a href="Hauptideal" title="Hauptideal">Hauptideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a42b43fe924fbe82623de1aa506862fc522c5c0c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.979ex; height:2.843ex;" alt="{\displaystyle (p)}" loading="lazy"> ein Primideal ist.<a href="#cite_note-2">[2]</a></li>

<li>Ein Ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}\subset R}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb087e68145dcae09dc05b48eaa0731ce289c70d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.025ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {p}}\subset R}" loading="lazy"> ist genau dann prim, wenn der <a href="Ringtheorie#Faktorring" class="mw-redirect" title="Ringtheorie">Faktorring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R/{\mathfrak {p}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9da6e6f1e63ba7c169bec572ae2ca16f1c58936.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.089ex; height:2.843ex;" alt="{\displaystyle R/{\mathfrak {p}}}" loading="lazy"> ein <a href="Integrit%C3%A4tsring" title="Integritätsring">Integritätsring</a> ist.</li>

<li>Enthält ein Primideal einen Durchschnitt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{1}\cap \ldots \cap {\mathfrak {a}}_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {a}}_{1}\cap \ldots \cap {\mathfrak {a}}_{n}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa3d77546d4caaf247682fc415ef814b9aa0cf0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.486ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {a}}_{1}\cap \ldots \cap {\mathfrak {a}}_{n}}" loading="lazy"> von endlich vielen Idealen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">, so enthält es auch eines der Ideale <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {a}}_{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e599f543e7b9df295f4407630f1a1123b8ac90e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.962ex; height:2.009ex;" alt="{\displaystyle {\mathfrak {a}}_{i}}" loading="lazy">.</li>

<li>Ein Ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}\subset R}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb087e68145dcae09dc05b48eaa0731ce289c70d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.025ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {p}}\subset R}" loading="lazy"> ist genau dann ein Primideal, wenn die Komplementärmenge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=R\setminus {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/870ea9871818daf44541ec52a278d897acd0652a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.719ex; height:2.843ex;" alt="{\displaystyle S=R\setminus {\mathfrak {p}}}" loading="lazy"> multiplikativ abgeschlossen ist. Das führt zum Begriff der <a href="Lokalisierung_(Algebra)" title="Lokalisierung (Algebra)">Lokalisierung</a> nach <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy">, worunter man den Ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{-1}R}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/36f81aa8006deb333465f99113727ab38fa80e04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.618ex; height:2.676ex;" alt="{\displaystyle S^{-1}R}" loading="lazy"> versteht, den man auch als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{\mathfrak {p}}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/124aee5f6d80492d2746e652dee942c36ef6e1c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.818ex; height:2.843ex;" alt="{\displaystyle R_{\mathfrak {p}}}" loading="lazy"> schreibt.<a href="#cite_note-3">[3]</a></li></ul>

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

<ul><li>Die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c112372c16743e8c02db5eac5c6de0659841bec6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle 2\mathbb {Z} }" loading="lazy"> der <a href="Parit%C3%A4t_(Mathematik)" title="Parität (Mathematik)">geraden</a> <a href="Ganze_Zahl" title="Ganze Zahl">ganzen Zahlen</a> ist ein Primideal im Ring <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"> der <a href="Ganze_Zahlen" class="mw-redirect" title="Ganze Zahlen">ganzen Zahlen</a>, da ein Produkt zweier ganzer Zahlen nur dann gerade ist, wenn wenigstens ein Faktor gerade ist.</li>

<li>Die Menge <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 6\mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 6\mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/94c7e28447bcf589a22415f04b96ff36e20074dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.713ex; height:2.176ex;" alt="{\displaystyle 6\mathbb {Z} }" loading="lazy"> der durch 6 teilbaren ganzen Zahlen ist kein Primideal in <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">, da 2·3 = 6 in der Teilmenge liegt, aber weder 2 noch 3.</li>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/400c9e7e65a49e58a0f80200299a8f3e0565a813.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.575ex; height:2.176ex;" alt="{\displaystyle R=2\mathbb {Z} }" loading="lazy"> ist das maximale Ideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}=4\mathbb {Z} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}=4\mathbb {Z} }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/38926d4d1055ceed84690bf09d76c6b5edd65011.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.594ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {m}}=4\mathbb {Z} }" loading="lazy"> kein Primideal.</li>

<li>Ein <a href="Maximales_Ideal" title="Maximales Ideal">maximales Ideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}\subseteq R}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3858513a1730cdfcb034d8996541d83ee95acf73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.645ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {m}}\subseteq R}" loading="lazy"> eines Ringes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> ist genau dann prim, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle RR\nsubseteq {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5075d078be3520f9310a38dab60f61c0569402ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.409ex; height:3.176ex;" alt="{\displaystyle RR\nsubseteq {\mathfrak {m}}}" loading="lazy">. Insbesondere ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {m}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/adc0e9162e96758157a34a6e44967288b481a7cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}" loading="lazy"> prim, falls <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 Einselement enthält.</li>

<li>Das <a href="Nullideal" class="mw-redirect" title="Nullideal">Nullideal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0)\subset R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/172cdd9b53fc6136885060392699a915f39c358b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.834ex; height:2.843ex;" alt="{\displaystyle (0)\subset R}" loading="lazy"> in einem kommutativen 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"> mit Einselement ist genau dann ein Primideal, 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"> ein <a href="Integrit%C3%A4tsring" title="Integritätsring">Integritätsbereich</a> ist. In einem nicht-kommutativen Ring gilt diese Äquivalenz nicht.</li>

<li>Das Urbild eines Primideals unter einem <a href="Ringhomomorphismus" title="Ringhomomorphismus">Ringhomomorphismus</a> kommutativer Ringe ist entweder der ganze Ring oder ein Primideal. Das gilt nicht allgemein, so ist etwa das Nullideal im Ring der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy">-Matrizen über einem Körper prim, aber dessen Urbild unter der Inklusion des Rings der (oberen oder unteren) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59d2b4cb72e304526cf5b5887147729ea259da78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}" loading="lazy">-Dreiecksmatrizen über dem Körper nicht.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Lying_Over_und_Going_Down">Lying Over und Going Down</h2></div>

Im Folgenden sei stets <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 kommutativer Ring und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\subset S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b11762f1e41a8b802a512ab53f3b2ece265856b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.362ex; height:2.176ex;" alt="{\displaystyle R\subset S}" loading="lazy"> eine <a href="Ganzes_Element" title="Ganzes Element">ganze Ringerweiterung</a>. Dann existiert zu jedem Primideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}\subset R}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bb087e68145dcae09dc05b48eaa0731ce289c70d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.025ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {p}}\subset R}" loading="lazy"> ein Primideal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {q}}\subset S}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3503f1c465de55c1e65c77af9adb06a4ccd4e8f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.735ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {q}}\subset S}" loading="lazy">, so dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {q}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79b0448036d3f2e513dd80d94095c45d81510269.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.137ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {q}}}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a14c125cdf81ac25d76edc2e8d557302c9f555a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {p}}}" loading="lazy"> liegt, d. h.

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f1ae3309cfdcfa83ab258aeef943e9d1d27c3bf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.744ex; height:2.676ex;" alt="{\displaystyle {\mathfrak {p}}={\mathfrak {q}}\cap R}" loading="lazy">.</dd></dl>

In diesem Fall sagt man auch, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S/R}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2223297d422eaf46aeca7b62cb8479918422b29b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.426ex; height:2.843ex;" alt="{\displaystyle S/R}" loading="lazy"> die Lying Over Eigenschaft erfüllt. Ist zudem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\colon R\hookrightarrow S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5fe784342258a656dfac737fb3c92f4d2ea3b05a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.482ex; height:2.509ex;" alt="{\displaystyle f\colon R\hookrightarrow S}" loading="lazy"> eine Einbettung 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"> in <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 die von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> induzierte Abbildung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{*}\!\colon \,\mathrm {Spec} (S)\longrightarrow \mathrm {Spec} (R)}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{*}\!\colon \,\mathrm {Spec} (S)\longrightarrow \mathrm {Spec} (R)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3053bc342e7a3210b5be374ed6dc64d9d78dd8dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.686ex; height:2.843ex;" alt="{\displaystyle f^{*}\!\colon \,\mathrm {Spec} (S)\longrightarrow \mathrm {Spec} (R)}" loading="lazy"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {q}}\longmapsto f^{-1}({\mathfrak {q}})}">

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/40d3544e0887045e3495c162ddcc89bc5f954acc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.832ex; height:3.176ex;" alt="{\displaystyle {\mathfrak {q}}\longmapsto f^{-1}({\mathfrak {q}})}" loading="lazy"> <a href="Surjektive_Funktion" title="Surjektive Funktion">surjektiv</a>.

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

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2223297d422eaf46aeca7b62cb8479918422b29b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.426ex; height:2.843ex;" alt="{\displaystyle S/R}" loading="lazy"> die Going Down Eigenschaft, falls folgendes gilt: Ist

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {p}}_{1}\supseteq {\mathfrak {p}}_{2}\supseteq \cdots \supseteq {\mathfrak {p}}_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90334db13bc165b36e1311c9609210f1e0e648a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.833ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {p}}_{1}\supseteq {\mathfrak {p}}_{2}\supseteq \cdots \supseteq {\mathfrak {p}}_{n}}" loading="lazy"></dd></dl>

eine Kette von Primidealen in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> und

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {q}}_{1}\supseteq {\mathfrak {q}}_{2}\supseteq \cdots \supseteq {\mathfrak {q}}_{m}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/502ecbb2c97b20cf6895d412fcfea239a2a5eba2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.213ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {q}}_{1}\supseteq {\mathfrak {q}}_{2}\supseteq \cdots \supseteq {\mathfrak {q}}_{m}}" loading="lazy"></dd></dl>

eine Kette von Primidealen in <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"> mit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m<n}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/490c01b0cb770144f28afd17bb5fef277daf6f38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.843ex;" alt="{\displaystyle m<n}" loading="lazy">, so dass außerdem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {q}}_{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a508b7b0dbf15d705b3f32de50aab601e44821e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.937ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {q}}_{i}}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}_{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebc456e8ce59c0b5f16401170ef68c996c25d512.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.962ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {p}}_{i}}" loading="lazy"> liegt für alle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq i\leq m}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be7492673ec3919cb5774b3ba4bb81a7c2365c88.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.202ex; height:2.343ex;" alt="{\displaystyle 1\leq i\leq m}" loading="lazy">, so lässt sich letztere zu einer Kette

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathfrak {q}}_{1}\supseteq {\mathfrak {q}}_{2}\supseteq \cdots \supseteq {\mathfrak {q}}_{n}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd80c659089228a4fb161996cc57a09e3eee6c2e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.756ex; height:2.509ex;" alt="{\displaystyle {\mathfrak {q}}_{1}\supseteq {\mathfrak {q}}_{2}\supseteq \cdots \supseteq {\mathfrak {q}}_{n}}" loading="lazy"></dd></dl>

ergänzen, so dass jedes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {q}}_{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5a508b7b0dbf15d705b3f32de50aab601e44821e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.937ex; height:2.176ex;" alt="{\displaystyle {\mathfrak {q}}_{i}}" loading="lazy"> über <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathfrak {p}}_{i}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebc456e8ce59c0b5f16401170ef68c996c25d512.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.962ex; height:2.343ex;" alt="{\displaystyle {\mathfrak {p}}_{i}}" loading="lazy"> liegt. Diese ist unter anderem dann erfüllt, wenn <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R,S}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/369b318eb1a5dcd7aceef09dd85d8641b3c1ac98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.297ex; height:2.509ex;" alt="{\displaystyle R,S}" loading="lazy"> <a href="Integrit%C3%A4tsring" title="Integritätsring">Integritätsringe</a> sind und <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"> ganzabgeschlossen ist.

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Louis H. Rowen: Ring Theory. Band 1. Academic Press Inc., Boston u. a. 1988, ISBN 0-125-99841-4 (Pure and Applied Mathematics 127), Definition 2.2.3'

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> K. Meyberg: Algebra, Teil 1, <a href="Carl_Hanser_Verlag" title="Carl Hanser Verlag">Carl Hanser Verlag</a> München (1975), ISBN 3-446-11965-5, Satz 3.6.5

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Ernst Kunz: Einführung in die kommutative Algebra und algebraische Geometrie, Vieweg (1980), ISBN 3-528-07246-6, Kapitel III, § 4, Beispiel d) hinter Satz 3.5

</li>

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