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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Funktionenring</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>Ein <b>Funktionenring</b> ist in der Mathematik (genauer der <a href="Ringtheorie" class="mw-redirect" title="Ringtheorie">Ringtheorie</a>) ein spezieller <a href="Ring_(Algebra)" title="Ring (Algebra)">Ring</a> von <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktionen</a>. Diese spielen eine große Rolle in der abstrakten <a href="Algebra" title="Algebra">Algebra</a>, <a href="Topologie_(Mathematik)" title="Topologie (Mathematik)">Topologie</a>, sowie zahlreichen Anwendungen der <a href="Mathematik" title="Mathematik">Mathematik</a> in Naturwissenschaften.
</p>
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>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 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 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> eine nichtleere <a href="Menge_(Mathematik)" title="Menge (Mathematik)">Menge</a> und
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} (M,R):=\{f\colon M\to R\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo>,</mo>
<mi>R</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>:<!-- : --></mo>
<mi>M</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>R</mi>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} (M,R):=\{f\colon M\to R\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f189469ca7d62384e362842dbd1e77c2cdf0b704.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.673ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} (M,R):=\{f\colon M\to R\}}" loading="lazy"></span></dd></dl>
<p>die Menge aller auf <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> definierten Funktionen mit Werten 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>.
Dann sind durch
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f+g)(x):=f(x)+g(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo>+</mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (f+g)(x):=f(x)+g(x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/30816ebf38289d507efbaaa8d096edbe36a76867.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.442ex; height:2.843ex;" alt="{\displaystyle (f+g)(x):=f(x)+g(x)}" loading="lazy"></span></dd>
<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 (f\cdot g)(x):=f(x)\cdot g(x)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>f</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi>g</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>:=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>⋅<!-- ⋅ --></mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (f\cdot g)(x):=f(x)\cdot g(x)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/02c9a6dc4b5686335efc577d86bf57bd2420a5a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.119ex; height:2.843ex;" alt="{\displaystyle (f\cdot g)(x):=f(x)\cdot g(x)}" loading="lazy"></span></dd></dl>
<p>Verknüpfungen erklärt, mit denen <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 {F} (M,R)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo>,</mo>
<mi>R</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} (M,R)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e43c2de95f15435467d6a7acbf3acc8fd3464d40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.47ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} (M,R)}" loading="lazy"></span> zu einem Ring wird, dem sogenannten Ring der Funktionen.
</p>
<div class="mw-heading mw-heading2"><h2 id="Wichtige_Eigenschaften">Wichtige Eigenschaften</h2></div>
<ul><li>Der 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 \mathbb {F} (M,R)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo>,</mo>
<mi>R</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} (M,R)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e43c2de95f15435467d6a7acbf3acc8fd3464d40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.47ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} (M,R)}" loading="lazy"></span> "ererbt" gewisse Eigenschaften 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 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>, wie etwa die <a href="Ringtheorie" class="mw-redirect" title="Ringtheorie">Kommutativität</a> und das <a href="Unit%C3%A4rer_Ring" class="mw-redirect" title="Unitärer Ring">Einselement</a>. Andere Eigenschaften, wie beispielsweise <a href="Nullteiler" title="Nullteiler">Nullteilerfreiheit</a>, werden nicht "vererbt".</li>
<li>Die Menge der konstanten Funktionen bildet einen zu <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="Ringhomomorphismus" title="Ringhomomorphismus">isomorphen</a> <a href="Unterring" class="mw-redirect" title="Unterring">Unterring</a> 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 F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span>. Damit kann <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> als Teilring 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 F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> betrachtet werden.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Beispiele">Beispiele</h2></div>
<ul><li>Wählt man 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> die Menge der reellen Zahlen <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 {R} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"></span> mit den üblichen Addition und Multiplikation und 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 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> eine <a href="Offene_Menge" title="Offene Menge">offene</a> Teilmenge 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 \mathbb {R} ^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"></span>, so kann man von <a href="Stetige_Funktion" title="Stetige Funktion">stetigen</a> beziehungsweise <a href="Differenzierbarkeit" title="Differenzierbarkeit">differenzierbaren</a> Funktionen sprechen. In diesem Falle sind die Mengen <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(M)=\{f\colon M\to R\mid f{\text{ ist stetig}}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>:<!-- : --></mo>
<mi>M</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>R</mi>
<mo>∣<!-- ∣ --></mo>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext> ist stetig</mtext>
</mrow>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C(M)=\{f\colon M\to R\mid f{\text{ ist stetig}}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/63dde0416ded59902508338129e1dcad35ac717b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.986ex; height:2.843ex;" alt="{\displaystyle C(M)=\{f\colon M\to R\mid f{\text{ ist stetig}}\}}" loading="lazy"></span> 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 D(M)=\{f\colon M\to R\mid f{\text{ ist differenzierbar}}\}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo fence="false" stretchy="false">{</mo>
<mi>f</mi>
<mo>:<!-- : --></mo>
<mi>M</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>R</mi>
<mo>∣<!-- ∣ --></mo>
<mi>f</mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext> ist differenzierbar</mtext>
</mrow>
<mo fence="false" stretchy="false">}</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D(M)=\{f\colon M\to R\mid f{\text{ ist differenzierbar}}\}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/66a8d3a24ae3b531da0a96430c94b627e5054d01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.199ex; height:2.843ex;" alt="{\displaystyle D(M)=\{f\colon M\to R\mid f{\text{ ist differenzierbar}}\}}" loading="lazy"></span> Unterringe 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 \mathbb {F} (M,R)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo>,</mo>
<mi>R</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} (M,R)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e43c2de95f15435467d6a7acbf3acc8fd3464d40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.47ex; height:2.843ex;" alt="{\displaystyle \mathbb {F} (M,R)}" loading="lazy"></span>. Dabei 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 D(M)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>D</mi>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle D(M)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a16bfd78fe6b7b8d2bc4d3911821e9f6d5316cff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.176ex; height:2.843ex;" alt="{\displaystyle D(M)}" loading="lazy"></span> ein Unterring 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 C(M)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C(M)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c99cfcfa6e76b435d982751f8237f16f44f603d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.018ex; height:2.843ex;" alt="{\displaystyle C(M)}" loading="lazy"></span>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Auswertungshomomorphismus">Auswertungshomomorphismus</h2></div>
<p>Für ein festes <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\in M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>∈<!-- ∈ --></mo>
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a\in M}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a85c91dfc793cef35c9e3a005e8ffb2bb20a46d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.513ex; height:2.176ex;" alt="{\displaystyle a\in M}" loading="lazy"></span> ist die Abbildung
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi \colon \mathbb {F} (M,R)\to R}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ϕ<!-- ϕ --></mi>
<mo>:<!-- : --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="double-struck">F</mi>
</mrow>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo>,</mo>
<mi>R</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">→<!-- → --></mo>
<mi>R</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \phi \colon \mathbb {F} (M,R)\to R}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e2ea47ecaa7db114bdffbef561d27078a338ec5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.267ex; height:2.843ex;" alt="{\displaystyle \phi \colon \mathbb {F} (M,R)\to R}" loading="lazy"></span></dd>
<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 f\mapsto f(a)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo stretchy="false">↦<!-- ↦ --></mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f\mapsto f(a)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/14a80bca3fe163e13eef6a492940e1c6aa38365b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.21ex; height:2.843ex;" alt="{\displaystyle f\mapsto f(a)}" loading="lazy"></span></dd></dl>
<p>ein <a href="Ringhomomorphismus" title="Ringhomomorphismus">Ringhomomorphismus</a>. Man bezeichnet ihn als Auswertungshomomorphismus oder auch einfach als die Auswertung an der Stelle <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\in M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>a</mi>
<mo>∈<!-- ∈ --></mo>
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle a\in M}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a85c91dfc793cef35c9e3a005e8ffb2bb20a46d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.513ex; height:2.176ex;" alt="{\displaystyle a\in M}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li><a href="Albrecht_Beutelspacher" title="Albrecht Beutelspacher">Albrecht Beutelspacher</a>: <i>Lineare Algebra. Eine Einführung in die Wissenschaft der Vektoren, Abbildungen und Matrizen.</i> 6. durchgesehene und ergänzte Auflage, Nachdruck. Vieweg + Teubner, Wiesbaden 2009, ISBN 978-3-528-56508-4 (<i>Mathematik für Studienanfänger</i>).</li>
<li><a href="Gerd_Fischer_(Mathematiker)" title="Gerd Fischer (Mathematiker)">Gerd Fischer</a>: <i>Lehrbuch der Algebra.</i> Vieweg, Wiesbaden 2008, ISBN 978-3-8348-0226-2 (<i>Vieweg Mathematik</i>).</li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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