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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Kommutative 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>Die <b>kommutative Algebra</b> ist das <a href="Teilgebiet_der_Mathematik" class="mw-redirect" title="Teilgebiet der Mathematik">Teilgebiet der Mathematik</a> im Bereich der <a href="Algebra" title="Algebra">Algebra</a>, das sich mit <a href="Kommutativer_Ring" class="mw-redirect" title="Kommutativer Ring">kommutativen Ringen</a> sowie deren <a href="Ideal_(Mathematik)" class="mw-redirect" title="Ideal (Mathematik)">Idealen</a>, <a href="Modul_(Mathematik)" title="Modul (Mathematik)">Moduln</a> und <a href="Algebra_(Struktur)" class="mw-redirect" title="Algebra (Struktur)">Algebren</a> befasst. Sie ist grundlegend für die Gebiete der <a href="Algebraische_Geometrie" title="Algebraische Geometrie">algebraischen Geometrie</a> und der <a href="Algebraische_Zahlentheorie" title="Algebraische Zahlentheorie">algebraischen Zahlentheorie</a>. Ein wichtiges Beispiel für kommutative Ringe sind <a href="Polynomring" title="Polynomring">Polynomringe</a>.
</p><p>Als Begründer der kommutativen Algebra kann man <a href="David_Hilbert" title="David Hilbert">David Hilbert</a> nennen.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Er scheint die <i>Idealtheorie</i> (so wurde die kommutative Algebra ursprünglich genannt) als alternativen Zugang zu zahlreichen Fragestellungen angesehen zu haben, der die damals dominierende <a href="Funktionentheorie" title="Funktionentheorie">Funktionentheorie</a> ablösen könnte. In diesem Zusammenhang waren ihm strukturelle Aspekte wichtiger als algorithmische; mit der wachsenden Leistungsfähigkeit von <a href="Computeralgebrasystem" title="Computeralgebrasystem">Computeralgebrasystemen</a> haben aber konkrete Berechnungen stark an Bedeutung innerhalb der kommutativen Algebra gewonnen.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Das Konzept der Moduln, das in Grundzügen auf <a href="Leopold_Kronecker" title="Leopold Kronecker">Leopold Kronecker</a> zurückgeht, verallgemeinert die Theorie der Ideale, die es als Spezialfall enthält. Diese Methoden wurden von <a href="Emmy_Noether" title="Emmy Noether">Emmy Noether</a> in die kommutative Algebra eingeführt und sind heute unverzichtbar.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>
</p><p>Die Theorie allgemeiner Ringe, die nicht kommutativ sein müssen, wird als nichtkommutative Algebra bezeichnet.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="Übliche_Annahmen"><span id=".C3.9Cbliche_Annahmen"></span>Übliche Annahmen</h2></div>
<p>In der kommutativen Algebra werden die Bezeichnungen <i>Modul, Ring</i> und <i>Algebra</i> üblicherweise in einem engeren Sinn benutzt:
</p>
<ul><li>Alle Moduln sind <a href="Modul_(Mathematik)#Moduln_über_einem_kommutativen_Ring_mit_Einselement" title="Modul (Mathematik)">unitär</a>: Wenn <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}">
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/92d98b82a3778f043108d4e20960a9193df57cbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}" loading="lazy"></span> das Einselement des Ringes ist, dann gilt für alle Elemente des Moduls:</li></ul>
<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 1\cdot m=m}">
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<annotation encoding="application/x-tex">{\displaystyle 1\cdot m=m}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07dbfdde4606dabc426a99b6db83ac54ca15f8cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.021ex; height:2.176ex;" alt="{\displaystyle 1\cdot m=m}" loading="lazy"></span></dd></dl>
<ul><li>Alle Ringe sind <a href="Unit%C3%A4rer_Ring" class="mw-redirect" title="Unitärer Ring">unitär</a> und <a href="Kommutativ" class="mw-redirect" title="Kommutativ">kommutativ</a>.</li>
<li><a href="Homomorphismus" title="Homomorphismus">Homomorphismen</a> zwischen <a href="Ring_(Algebra)" title="Ring (Algebra)">Ringen</a> bilden Einselemente auf Einselemente ab.</li>
<li>Ein Unterring hat dasselbe Einselement wie der Oberring.</li>
<li>Alle Algebren sind unitär, kommutativ und <a href="Assoziativgesetz" title="Assoziativgesetz">assoziativ</a>.</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Skripte">Skripte</h3></div>
<ul><li>Andreas Gathmann: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Commutative Algebra</cite>. 2014 (englisch, <a rel="nofollow" class="external text" href="https://www.mathematik.uni-kl.de/~gathmann/class/commalg-2013/commalg-2013.pdf">uni-kl.de</a> [PDF]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kommutative+Algebra&rft.au=Andreas+Gathmann&rft.btitle=Commutative+Algebra&rft.date=2014&rft.genre=book" style="display:none"> </span></li>
<li>Dieter Neßelmann: <cite style="font-style:italic">Ringe und Moduln</cite>. 2005 (<a rel="nofollow" class="external text" href="http://www.math.uni-rostock.de/~nesselmann/RingeModuln/Ringe_Moduln.pdf">uni-rostock.de</a> [PDF]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kommutative+Algebra&rft.au=Dieter+Ne%C3%9Felmann&rft.btitle=Ringe+und+Moduln&rft.date=2005&rft.genre=book" style="display:none"> </span></li>
<li>Pawel Sosna: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Commutative Algebra</cite>. 2015 (englisch, <a rel="nofollow" class="external text" href="https://www.math.uni-hamburg.de/home/sosna/commalg/commalgebra.pdf">uni-hamburg.de</a> [PDF]).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kommutative+Algebra&rft.au=Pawel+Sosna&rft.btitle=Commutative+Algebra&rft.date=2015&rft.genre=book" style="display:none"> </span></li></ul>
<div class="mw-heading mw-heading3"><h3 id="Lehrwerke_&_Monografien"><span id="Lehrwerke_.26_Monografien"></span>Lehrwerke & Monografien</h3></div>
<ul><li><a href="Oscar_Zariski" title="Oscar Zariski">Oscar Zariski</a>, <a href="Pierre_Samuel" title="Pierre Samuel">Pierre Samuel</a>: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Commutative Algebra Vol. I</cite> (= <cite class="lang" lang="en" dir="auto" style="font-style:italic">Graduate Texts in Mathematics</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>28</span>). <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>28</span>. Springer Berlin Heidelberg, Berlin, Heidelberg 1975, ISBN 978-0-387-90089-6 (englisch, <a rel="nofollow" class="external text" href="https://link.springer.com/book/9780387900896">springer.com</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kommutative+Algebra&rft.au=Oscar+Zariski%2C+Pierre+Samuel&rft.btitle=Commutative+Algebra+Vol.+I&rft.date=1975&rft.genre=book&rft.isbn=9780387900896&rft.place=Berlin%2C+Heidelberg&rft.pub=Springer+Berlin+Heidelberg&rft.series=Graduate+Texts+in+Mathematics&rft.volume=28" style="display:none"> </span></li>
<li><a href="Oscar_Zariski" title="Oscar Zariski">Oscar Zariski</a>, <a href="Pierre_Samuel" title="Pierre Samuel">Pierre Samuel</a>: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Commutative Algebra Vol. II</cite> (= <cite class="lang" lang="en" dir="auto" style="font-style:italic">Graduate Texts in Mathematics</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>29</span>). <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>29</span>. Springer Berlin Heidelberg, Berlin, Heidelberg 1960, ISBN 978-3-662-27753-9, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-662-29244-0">10.1007/978-3-662-29244-0</a></span> (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kommutative+Algebra&rft.au=Oscar+Zariski%2C+Pierre+Samuel&rft.btitle=Commutative+Algebra+Vol.+II&rft.date=1960&rft.doi=10.1007%2F978-3-662-29244-0&rft.genre=book&rft.isbn=9783662277539&rft.place=Berlin%2C+Heidelberg&rft.pub=Springer+Berlin+Heidelberg&rft.series=Graduate+Texts+in+Mathematics&rft.volume=29" style="display:none"> </span></li>
<li><a href="Michael_Francis_Atiyah" title="Michael Francis Atiyah">Michael F. Atiyah</a>, <a href="Ian_Macdonald" title="Ian Macdonald">Ian G. MacDonald</a>: <i>Introduction to Commutative Algebra.</i> Addison-Wesley, Reading MA 1969, ISBN 0-201-00361-9.</li>
<li>Winfried Bruns, <a href="J%C3%BCrgen_Herzog_(Mathematiker)" title="Jürgen Herzog (Mathematiker)">Jürgen Herzog</a>: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Cohen-Macaulay rings</cite> (= <cite class="lang" lang="en" dir="auto" style="font-style:italic">Cambridge studies in advanced mathematics</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>39</span>). Cambridge University Press, 1993 (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kommutative+Algebra&rft.au=Winfried+Bruns%2C+J%C3%BCrgen+Herzog&rft.btitle=Cohen-Macaulay+rings&rft.date=1993&rft.genre=book&rft.pub=Cambridge+University+Press&rft.series=Cambridge+studies+in+advanced+mathematics" style="display:none"> </span></li>
<li><a href="David_Eisenbud" title="David Eisenbud">David Eisenbud</a>: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Commutative Algebra</cite> (= <cite class="lang" lang="en" dir="auto" style="font-style:italic">Graduate Texts in Mathematics</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>150</span>). Springer New York, New York, NY 1995, ISBN 978-3-540-78122-6, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-1-4612-5350-1">10.1007/978-1-4612-5350-1</a></span> (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kommutative+Algebra&rft.au=David+Eisenbud&rft.btitle=Commutative+Algebra&rft.date=1995&rft.doi=10.1007%2F978-1-4612-5350-1&rft.genre=book&rft.isbn=9783540781226&rft.place=New+York%2C+NY&rft.pub=Springer+New+York&rft.series=Graduate+Texts+in+Mathematics" style="display:none"> </span></li>
<li>Jürgen Böhm: <cite style="font-style:italic">Kommutative Algebra und Algebraische Geometrie</cite>. Springer Berlin Heidelberg, Berlin, Heidelberg 2019, ISBN 978-3-662-59481-0, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-662-59482-7">10.1007/978-3-662-59482-7</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kommutative+Algebra&rft.au=J%C3%BCrgen+B%C3%B6hm&rft.btitle=Kommutative+Algebra+und+Algebraische+Geometrie&rft.date=2019&rft.doi=10.1007%2F978-3-662-59482-7&rft.genre=book&rft.isbn=9783662594810&rft.place=Berlin%2C+Heidelberg&rft.pub=Springer+Berlin+Heidelberg" style="display:none"> </span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text"><a href="David_Eisenbud" title="David Eisenbud">David Eisenbud</a>: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Commutative Algebra</cite>. Springer Science+Business Media, New York 1995, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-1-4612-5350-1">10.1007/978-1-4612-5350-1</a></span> (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kommutative+Algebra&rft.au=David+Eisenbud&rft.btitle=Commutative+Algebra&rft.date=1995&rft.doi=10.1007%2F978-1-4612-5350-1&rft.genre=book&rft.place=New+York&rft.pub=Springer+Science%2BBusiness+Media" style="display:none"> </span></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a href="Otto_Toeplitz" title="Otto Toeplitz">Otto Toeplitz</a>: <cite style="font-style:italic">Der Algebraiker Hilbert</cite>. In: <cite style="font-style:italic">Die Naturwissenschaften</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>10</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>4</span>, Januar 1922, <a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%220028-1042%22&key=cql">0028-1042</a></span>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>73–77</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/BF01591616">10.1007/BF01591616</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Kommutative+Algebra&rft.atitle=Der+Algebraiker+Hilbert&rft.au=Otto+Toeplitz&rft.date=1922-01&rft.doi=10.1007%2FBF01591616&rft.genre=journal&rft.issn=0028-1042&rft.issue=4&rft.jtitle=Die+Naturwissenschaften&rft.pages=73-77&rft.volume=10" style="display:none"> </span></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text"><a href="Martin_Kreuzer" title="Martin Kreuzer">Martin Kreuzer</a>, Lorenzo Robbiano: <cite style="font-style:italic">Computational Linear and Commutative Algebra</cite>. Springer International Publishing, Cham 2016, ISBN 978-3-319-43599-2, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-319-43601-2">10.1007/978-3-319-43601-2</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kommutative+Algebra&rft.au=Martin+Kreuzer%2C+Lorenzo+Robbiano&rft.btitle=Computational+Linear+and+Commutative+Algebra&rft.date=2016&rft.doi=10.1007%2F978-3-319-43601-2&rft.genre=book&rft.isbn=9783319435992&rft.place=Cham&rft.pub=Springer+International+Publishing" style="display:none"> </span></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text"><span class="cite">Noémie Combe: <a rel="nofollow" class="external text" href="https://www.mpg.de/16548098/emmy-noether"><i>"Without Emmy Noether, there would be a huge gap in mathematics and its understanding".</i></a> <a href="Max-Planck-Gesellschaft" title="Max-Planck-Gesellschaft">Max-Planck-Gesellschaft</a>,<span class="Abrufdatum"> abgerufen am 28. Oktober 2022</span> (englisch).</span><span style="display: none;" class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rfr_id=info%3Asid%2Fde.wikipedia.org%3AKommutative+Algebra&rft.title=%22Without+Emmy+Noether%2C+there+would+be+a+huge+gap+in+mathematics+and+its+understanding%22&rft.description=%22Without+Emmy+Noether%2C+there+would+be+a+huge+gap+in+mathematics+and+its+understanding%22&rft.identifier=https%3A%2F%2Fwww.mpg.de%2F16548098%2Femmy-noether&rft.creator=No%C3%A9mie+Combe&rft.publisher=%5B%5BMax-Planck-Gesellschaft%5D%5D&rft.language=en"> </span></span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Jeremy Gray: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Emmy Noether</cite>. In: <cite class="lang" lang="en" dir="auto" style="font-style:italic">A History of Abstract Algebra</cite>. Springer International Publishing, Cham 2018, ISBN 978-3-319-94772-3, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>289–295</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-3-319-94773-0_28">10.1007/978-3-319-94773-0_28</a></span> (englisch).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kommutative+Algebra&rft.atitle=Emmy+Noether&rft.au=Jeremy+Gray&rft.btitle=A+History+of+Abstract+Algebra&rft.date=2018&rft.doi=10.1007%2F978-3-319-94773-0_28&rft.genre=book&rft.isbn=9783319947723&rft.pages=289-295&rft.place=Cham&rft.pub=Springer+International+Publishing" style="display:none"> </span></span>
</li>
<li id="cite_note-6"><span class="mw-cite-backlink"><a href="#cite_ref-6">↑</a></span> <span class="reference-text"><a href="Benson_Farb" title="Benson Farb">Benson Farb</a>, R. Keith Dennis: <cite style="font-style:italic">Noncommutative Algebra</cite> (= <cite style="font-style:italic">Graduate Texts in Mathematics</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>144</span>). Springer New York, New York, NY 1993, ISBN 978-1-4612-6936-6, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1007/978-1-4612-0889-1">10.1007/978-1-4612-0889-1</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Kommutative+Algebra&rft.au=Benson+Farb%2C+R.+Keith+Dennis&rft.btitle=Noncommutative+Algebra&rft.date=1993&rft.doi=10.1007%2F978-1-4612-0889-1&rft.genre=book&rft.isbn=9781461269366&rft.place=New+York%2C+NY&rft.pub=Springer+New+York&rft.series=Graduate+Texts+in+Mathematics" style="display:none"> </span></span>
</li>
</ol>
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