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<title>Primelement</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Primelement</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>Der Begriff <b>Primelement</b> ist in der <a href="Kommutative_Algebra" title="Kommutative Algebra">kommutativen Algebra</a> eine Verallgemeinerung des Begriffs der <a href="Primzahl" title="Primzahl">Primzahl</a> auf <i>kommutative unitäre <a href="Ring_(Algebra)" title="Ring (Algebra)">Ringe</a></i>.

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

<p>Ein Element <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 c}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> eines kommutativen unitären Ringes <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,+,\cdot ,0,1)}">

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<annotation encoding="application/x-tex">{\displaystyle (R,+,\cdot ,0,1)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/01dcb973231d7d7d0e2fcd7b069572ccc3a1c714.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.489ex; height:2.843ex;" alt="{\displaystyle (R,+,\cdot ,0,1)}" loading="lazy"></span> heißt <i>Primelement</i>, falls <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 c}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> weder 0 noch eine <a href="Einheit_(Mathematik)" title="Einheit (Mathematik)">Einheit</a> ist und für alle <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 a,b\in R}">

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

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</math></span><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"></span> gilt: <a href="Teilbarkeit" title="Teilbarkeit">Teilt</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 c}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> das Produkt <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 a\cdot b}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/620419d3ed53abc98659a5fc0f3a5eb6177830ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.906ex; height:2.176ex;" alt="{\displaystyle a\cdot b}" loading="lazy"></span>, dann teilt <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 c}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> auch <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 a}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> oder <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 b}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy"></span>.

</p><p>In Symbolnotation: <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 c{\mbox{ ist prim }}\Leftrightarrow \ c\neq 0\ \land \ c\nmid 1\ \land \ \forall a,b\in R:\ c\mid (a\cdot b)\Rightarrow (c\mid a)\lor (c\mid b).}">

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<annotation encoding="application/x-tex">{\displaystyle c{\mbox{ ist prim }}\Leftrightarrow \ c\neq 0\ \land \ c\nmid 1\ \land \ \forall a,b\in R:\ c\mid (a\cdot b)\Rightarrow (c\mid a)\lor (c\mid b).}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe0f769b4b03c74e6411ff5c5420ce895bb4480a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:69.974ex; height:2.843ex;" alt="{\displaystyle c{\mbox{ ist prim }}\Leftrightarrow \ c\neq 0\ \land \ c\nmid 1\ \land \ \forall a,b\in R:\ c\mid (a\cdot b)\Rightarrow (c\mid a)\lor (c\mid b).}" loading="lazy"></span>

</p><p>Primelemente sind also diejenigen Elemente abgesehen von 0 und Einheiten, die, wenn sie in irgendeinem Produkt vorkommen, auch in mindestens einem der Faktoren vorkommen.<sup id="cite_ref-Bosch201_1-0" class="reference"><a href="#cite_note-Bosch201-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

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<div class="mw-heading mw-heading2"><h2 id="Irreduzible_Elemente">Irreduzible Elemente</h2></div>

<p>Eine andere Verallgemeinerung des Primzahlbegriffs sind <a href="Ringtheorie#Irreduzibilität" class="mw-redirect" title="Ringtheorie">irreduzible</a> Elemente, die dadurch definiert sind, dass sie keine Einheiten sind und nicht als Produkt von zwei Nicht-Einheiten dargestellt werden können. Im Allgemeinen ist weder jedes Primelement irreduzibel noch jedes irreduzible Element prim (siehe <a href="#Beispiele">Beispiele</a>). Aber in einem <a href="Integrit%C3%A4tsring" title="Integritätsring">Integritätsring</a> ist jedes Primelement irreduzibel, und in einem <a href="Faktorieller_Ring" title="Faktorieller Ring">faktoriellen Ring</a> ist auch umgekehrt jedes irreduzible Element prim.

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<div class="mw-heading mw-heading2"><h2 id="Sätze_über_Primelemente"><span id="S.C3.A4tze_.C3.BCber_Primelemente"></span>Sätze über Primelemente</h2></div>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> ein Primelement 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 e}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"></span> eine Einheit, 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 c\cdot e}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3310e14c2d8d737d164f5552ea8e376efa7c0073.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.769ex; height:1.676ex;" alt="{\displaystyle c\cdot e}" loading="lazy"></span> ebenfalls ein Primelement.</li>

<li>Eine Nichteinheit <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 c\neq 0}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/be27396bd0e62003728d08329a8767eee94409e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.268ex; height:2.676ex;" alt="{\displaystyle c\neq 0}" loading="lazy"></span> ist genau dann ein Primelement, wenn das <a href="Hauptideal" title="Hauptideal">Hauptideal</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 (c)}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0cced16c6169d8653fa3512190318eb26be63d8e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.816ex; height:2.843ex;" alt="{\displaystyle (c)}" loading="lazy"></span> ein <a href="Primideal" title="Primideal">Primideal</a> ist.</li>

<li>Ein <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</a> besteht nur aus der Null und Einheiten und enthält somit keine Primelemente.</li>

<li>In einem faktoriellen Ring lässt sich jedes Element außer 0 bis auf Einheitsfaktoren und Reihenfolge eindeutig als Produkt von Primelementen darstellen.</li></ul>

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

<ul><li>Die Primelemente im Ring der <a href="Ganze_Zahl" title="Ganze Zahl">ganzen Zahlen</a> sind genau die <a href="Primzahl" title="Primzahl">Primzahlen</a> (2, 3, 5, 7, 11, …) und ihre <a href="Gegenzahl" class="mw-redirect" title="Gegenzahl">Gegenzahlen</a> (−2, −3, −5, −7, −11, …).</li>

<li>Die Primelemente im Ring der <a href="Gau%C3%9Fsche_Zahl" title="Gaußsche Zahl">Gaußschen Zahlen</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 \mathbb {Z} [i]}">

</mrow>

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9ffa94e9e2e6d9e5e5373d5fafb954b902743fde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.646ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} [i]}" loading="lazy"></span> sind bis auf die Einheitsfaktoren <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 \pm 1,\pm i}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab0b1c6c9b0d9c23f32d1a856110c08c9be255c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.615ex; height:2.509ex;" alt="{\displaystyle \pm 1,\pm i}" loading="lazy"></span> genau die Primzahlen der Form <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 4k+3,\ k\in \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4k+3,\ k\in \mathbb {Z} }</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/287d8587646c065fa01fb0caf41a14ce796a0554.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.593ex; height:2.509ex;" alt="{\displaystyle 4k+3,\ k\in \mathbb {Z} }" loading="lazy"></span> und die Elemente <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 a+b\cdot i,\ a,b\in \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/368a7355c60084834b1c141aa01b3370122f7593.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.816ex; height:2.509ex;" alt="{\displaystyle a+b\cdot i,\ a,b\in \mathbb {Z} }" loading="lazy"></span>, für die <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 a^{2}+b^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a^{2}+b^{2}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14fc28103c5d2aa9276728469f82c9f415f4b257.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.176ex; height:2.843ex;" alt="{\displaystyle a^{2}+b^{2}}" loading="lazy"></span> eine Primzahl ist, also sind beispielsweise <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 3,\,7,\,11,\,1+i,\,2+3i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3,\,7,\,11,\,1+i,\,2+3i}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b251f5e3f1c345eec6741753ecc54fb2b361df2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:21.107ex; height:2.509ex;" alt="{\displaystyle 3,\,7,\,11,\,1+i,\,2+3i}" loading="lazy"></span> Primelemente, nicht aber <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 2=(1+i)\cdot (1-i)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0798322132eb66ac0dfe6c0527c792ad9711a050.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.169ex; height:2.843ex;" alt="{\displaystyle 2=(1+i)\cdot (1-i)}" loading="lazy"></span>, <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 5=(2+i)\cdot (2-i)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 5=(2+i)\cdot (2-i)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/381954c260b13eb7c9c4f72bdf3ac310795984d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.169ex; height:2.843ex;" alt="{\displaystyle 5=(2+i)\cdot (2-i)}" loading="lazy"></span> oder <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 3+i=(1+i)\cdot (2-i)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 3+i=(1+i)\cdot (2-i)}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3575f4582cbdadfa3cb7f449a882164f432303ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.812ex; height:2.843ex;" alt="{\displaystyle 3+i=(1+i)\cdot (2-i)}" loading="lazy"></span> (zum Beweis siehe <a href="Pierre_de_Fermat" title="Pierre de Fermat">Fermats</a> <a href="Zwei-Quadrate-Satz" title="Zwei-Quadrate-Satz">Zwei-Quadrate-Satz</a>).</li>

<li>Im Integritätsring <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} [i{\sqrt {5}}]}">

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} [i{\sqrt {5}}]}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/00d2f0f809f3e7ad192658c61cc1ae46e2a1901d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.745ex; height:3.009ex;" alt="{\displaystyle \mathbb {Z} [i{\sqrt {5}}]}" loading="lazy"></span> (enthält alle Zahlen der Form <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 a+b\cdot i{\sqrt {5}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a+b\cdot i{\sqrt {5}}}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73daefa44e98c28dedf389861430331325f92b26.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.648ex; height:2.843ex;" alt="{\displaystyle a+b\cdot i{\sqrt {5}}}" loading="lazy"></span> mit <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 a,b\in \mathbb {Z} }">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6271473ca60d1b6878a1430d12324319bebe7ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.652ex; height:2.509ex;" alt="{\displaystyle a,b\in \mathbb {Z} }" loading="lazy"></span>) ist die Zahl 2 irreduzibel, aber nicht prim. Das ist so, weil 6 zwar von 2 geteilt wird, sich aber als Produkt <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 (1+i{\sqrt {5}})\cdot (1-i{\sqrt {5}})}">

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1+i{\sqrt {5}})\cdot (1-i{\sqrt {5}})}</annotation>

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bcc73806955a237badb3e61687a5b7e2a7d60239.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.105ex; height:3.009ex;" alt="{\displaystyle (1+i{\sqrt {5}})\cdot (1-i{\sqrt {5}})}" loading="lazy"></span> schreiben lässt, wobei keiner der Faktoren durch 2 teilbar ist.</li>

<li>Im <a href="Direktes_Produkt#Direktes_Produkt_von_Ringen,_Vektorräumen_und_Moduln" title="Direktes Produkt">Produktring</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 \mathbb {Z} \times \mathbb {Z} }">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/463bb631669181cf8a8950ac822ada5eb0542ca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.941ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} \times \mathbb {Z} }" loading="lazy"></span> 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 (1,0)=(1,0)\cdot (1,0)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/039deb5d3efd64b4bd633018ba0e0c346d46b55d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.282ex; height:2.843ex;" alt="{\displaystyle (1,0)=(1,0)\cdot (1,0)}" loading="lazy"></span> ein Primelement, das nicht irreduzibel ist.</li></ul>

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

<li id="cite_note-Bosch201-1"><span class="mw-cite-backlink"><a href="#cite_ref-Bosch201_1-0">↑</a></span> <span class="reference-text"><a href="Siegfried_Bosch" title="Siegfried Bosch">Siegfried Bosch</a>: <i>Algebra.</i> 7. Auflage 2009, Springer-Verlag, ISBN 3-540-40388-4, S. 201.</span>

</li>

</ol>

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

<div class="sisterproject" style="margin:0.1em 0 0 0;"><span class="noviewer" style="display:inline-block; line-height:10px; min-width:1.6em; text-align:center;" aria-hidden="true" role="presentation"><span class="mw-default-size" typeof="mw:File"><span title="Wiktionary"></span></span></span><b><a href="https://de.wiktionary.org/wiki/Primelement" class="extiw external" title="wikt:Primelement">Wiktionary: Primelement</a></b> – Bedeutungserklärungen, Wortherkunft, Synonyme, Übersetzungen</div></div><div><div class="zim-footer">

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