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<title>Assoziative Algebra</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Assoziative Algebra</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><b>Assoziative Algebra</b> ist ein Begriff aus der <a href="Abstrakte_Algebra" title="Abstrakte Algebra">abstrakten Algebra</a>, einem Teilgebiet der <a href="Mathematik" title="Mathematik">Mathematik</a>. Es handelt sich um eine <a href="Algebraische_Struktur" title="Algebraische Struktur">algebraische Struktur</a>, die den Begriff des <a href="Vektorraum" title="Vektorraum">Vektorraums</a> bzw. des <a href="Modul_(Mathematik)" title="Modul (Mathematik)">Moduls</a> dahingehend erweitert, dass zusätzlich zur Vektoraddition eine <a href="Assoziativgesetz" title="Assoziativgesetz">assoziative</a> Multiplikation als <a href="Innere_Verkn%C3%BCpfung" class="mw-redirect" title="Innere Verknüpfung">innere Verknüpfung</a> definiert wird.

</p><p>In anderen Worten, eine assoziative Algebra ist eine <a href="Algebra_%C3%BCber_einem_K%C3%B6rper" title="Algebra über einem Körper"><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 K}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span>-Algebra</a> respektive <a href="Algebra_%C3%BCber_einem_kommutativen_Ring" title="Algebra über einem kommutativen Ring"><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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

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</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>-Algebra</a>, deren Multiplikation zusätzlich assoziativ ist.

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

<p>Ein Vektorraum <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>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> über einem <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</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 K}">

</mstyle>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> oder ein Modul <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_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> über einem <a href="Ring_(Algebra)" title="Ring (Algebra)">Ring</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 R}">

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

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</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> zusammen mit einer <a href="Bilineare_Abbildung" title="Bilineare Abbildung">bilinearen Abbildung</a> genannt Multiplikation

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<annotation encoding="application/x-tex">{\displaystyle \cdot \colon A\times A\longrightarrow A,\quad (a,b)\longmapsto a\cdot b}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6bff5f870708cddc237013424dbbb0935a90a176.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.276ex; height:2.843ex;" alt="{\displaystyle \cdot \colon A\times A\longrightarrow A,\quad (a,b)\longmapsto a\cdot b}" loading="lazy"></span></dd></dl>

<p>heißt <b>assoziative Algebra</b>, wenn 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,c\in A}">

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a4de91bb40e7fb8dc8e5b35c93d556ea584986da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.886ex; height:2.509ex;" alt="{\displaystyle a,b,c\in A}" loading="lazy"></span> das folgende Assoziativgesetz gilt:

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d64ab192a7406fc3c3679bcce506e5e6cc7dd2f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.549ex; height:2.843ex;" alt="{\displaystyle a\cdot (b\cdot c)=(a\cdot b)\cdot c.}" loading="lazy"></span></dd></dl>

<p>Es handelt sich also um eine spezielle <a href="Algebra_%C3%BCber_einem_K%C3%B6rper" title="Algebra über einem Körper">Algebra über einem Körper</a> oder eine spezielle <a href="Algebra_%C3%BCber_einem_kommutativen_Ring" title="Algebra über einem kommutativen Ring">Algebra über einem kommutativen Ring</a>.

</p><p>Wenn die Multiplikation zusätzlich <a href="Kommutativgesetz" title="Kommutativgesetz">kommutativ</a> ist, dann spricht man von einer <b>kommutativen assoziativen Algebra</b>.

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<div class="mw-heading mw-heading3"><h3 id="Erläuterungen"><span id="Erl.C3.A4uterungen"></span>Erläuterungen</h3></div>

<p>Der Vektorraum respektive der Modul <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>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span>, der eine <i>assoziative Algebra</i> ist, besitzt nun neben

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<ul><li>der Addition: <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 A}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72e41c8d4ebf5e73c3812701fda79543122dec08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.652ex; height:2.343ex;" alt="{\displaystyle a+b\in A}" loading="lazy"></span> für <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 A}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/923af80638e57592b33fa715fb6c23651729fb73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.845ex; height:2.509ex;" alt="{\displaystyle a,b\in A}" loading="lazy"></span>,</li>

<li>der <a href="Skalarmultiplikation" title="Skalarmultiplikation">Skalarmultiplikation</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 s\cdot a\in A}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f407275e28e7e0a1ede0009541cb43ef56fa456b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.583ex; height:2.176ex;" alt="{\displaystyle s\cdot a\in A}" loading="lazy"></span> für <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 A}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a97387981adb5d65f74518e20b6785a284d7abd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.814ex; height:2.176ex;" alt="{\displaystyle a\in A}" 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 s\in K}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd087ce0dc9a6f7ae49d660b19ff5842d5f3c491.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.997ex; height:2.176ex;" alt="{\displaystyle s\in K}" loading="lazy"></span> respektive im Falle des Moduls <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 s\in R}">

</mstyle>

</mrow>

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

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cbe8fabbb611503ed849efa948775b1488903ff1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.695ex; height:2.176ex;" alt="{\displaystyle s\in R}" loading="lazy"></span>,</li></ul>

<p>eine zusätzliche Operation

</p>

<ul><li>die Multiplikation <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\in A}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d133efd3d1866929379b82c8ad580c47a7bde4d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.49ex; height:2.176ex;" alt="{\displaystyle a\cdot b\in A}" loading="lazy"></span> für <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 A}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/923af80638e57592b33fa715fb6c23651729fb73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.845ex; height:2.509ex;" alt="{\displaystyle a,b\in A}" loading="lazy"></span>,</li></ul>

<p>welche dem Assoziativgesetz unterliegt.

</p><p>Der Raum <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>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> zusammen mit der Addition und der Skalarmultiplikation bildet die Vektorraum-Struktur <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 _{s})}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (A,+,\cdot _{s})}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2ffbe8c70f084e0792cd5fa92cc834d6a2ecb423.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.079ex; height:2.843ex;" alt="{\displaystyle (A,+,\cdot _{s})}" loading="lazy"></span>, während <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>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span> zusammen mit der Addition und der Multiplikation die Ring-Struktur <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 )}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/933ada95d5707c0af8a812755d316c22280dc4a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.075ex; height:2.843ex;" alt="{\displaystyle (A,+,\cdot )}" loading="lazy"></span> bildet (wobei hier <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 \cdot _{s}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/856eb9397534975c366ba27c27bb9c888a9a1a35.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.65ex; height:1.676ex;" alt="{\displaystyle \cdot _{s}}" loading="lazy"></span> die Skalarmultiplikation sein soll).

</p>

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

<ul><li>Die Menge aller <a href="Polynom" title="Polynom">Polynome</a> mit Koeffizienten aus einem <a href="K%C3%B6rper_(Algebra)" title="Körper (Algebra)">Körper</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 K}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> bilden (mit der üblichen Multiplikation) eine assoziative Algebra über diesem Körper.</li>

<li>Die <a href="Endomorphismus" title="Endomorphismus">Endomorphismen</a> eines Vektorraums <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> bilden mit der Verkettung eine assoziative Algebra. Hierbei ist die Verknüpfung <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 *}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e9972f426d9e07855984f73ee195a21dbc21755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" loading="lazy"></span> nicht kommutativ, sofern die Dimension 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 V}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> größer als 1 ist.</li>

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span> ein unendlichdimensionaler Vektorraum und betrachtet man nur die Endomorphismen mit endlich-dimensionalem Bild, erhält man ein Beispiel, bei dem <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 *}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8e9972f426d9e07855984f73ee195a21dbc21755.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: 0.079ex; margin-bottom: -0.25ex; width:1.162ex; height:1.509ex;" alt="{\displaystyle *}" loading="lazy"></span> kein Einselement hat.</li>

<li>Der Vektorraum aller reell- oder komplexwertigen <a href="Funktion_(Mathematik)" title="Funktion (Mathematik)">Funktionen</a> auf einem beliebigen topologischem Raum bildet eine assoziative Algebra; dabei werden die Funktionen punktweise addiert und multipliziert.</li>

<li>Der Vektorraum aller <a href="Stetig" class="mw-redirect" title="Stetig">stetigen</a> reell- oder <a href="Komplexwertig" class="mw-redirect" title="Komplexwertig">komplexwertigen</a> Funktionen auf einem <a href="Banachraum" title="Banachraum">Banachraum</a> bildet eine assoziative Algebra, bzw. sogar eine <a href="Banach-Algebra" class="mw-redirect" title="Banach-Algebra">Banach-Algebra</a>.</li>

<li>Der <a href="Matrizenraum" title="Matrizenraum">Matrizenraum</a> aller <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (n\times n)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b0e369b52ee16c33d83f7cd0eb0f562fd91b7f3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.439ex; height:2.843ex;" alt="{\displaystyle (n\times n)}" loading="lazy"></span>-<a href="Matrix_(Mathematik)" title="Matrix (Mathematik)">Matrizen</a> bildet zusammen mit der <a href="Matrizenmultiplikation" title="Matrizenmultiplikation">Matrizenmultiplikation</a> eine assoziative Algebra.</li>

<li>Die komplexen Zahlen bilden eine assoziative Algebra über dem Körper der <a href="Reelle_Zahl" title="Reelle Zahl">reellen Zahlen</a>.</li>

<li>Die <a href="Quaternionen" class="mw-redirect" title="Quaternionen">Quaternionen</a> sind eine assoziative Algebra über dem Körper der reellen Zahlen, aber nicht über den komplexen Zahlen.</li>

<li>Die <a href="Weyl-Algebra" title="Weyl-Algebra">Weyl-Algebren</a></li></ul>

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

<ul><li><a href="Serge_Lang" title="Serge Lang">Serge Lang</a>: <i>Algebra.</i> Revised 3rd Edition, Springer-Verlag, 2002, ISBN 0-387-95385-X</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2025-07-27" href="https://de.wikipedia.org/wiki/?title=Assoziative_Algebra&oldid=258333070">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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