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<div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" 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href="https://cv.wikipedia.org/wiki/%C4%94%D1%81%D0%BA%D0%B5%D1%80_(%D0%B0%D0%BB%D0%B3%D0%B5%D0%B1%D1%80%D0%B0)" title="Ĕскер (алгебра) – Chuvash" lang="cv" hreflang="cv" data-title="Ĕскер (алгебра)" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/T%C4%9Bleso_(algebra)" title="Těleso (algebra) – Czech" lang="cs" hreflang="cs" data-title="Těleso (algebra)" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Schiefk%C3%B6rper" title="Schiefkörper – German" lang="de" hreflang="de" data-title="Schiefkörper" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Kaldkorpus" title="Kaldkorpus – Estonian" lang="et" hreflang="et" data-title="Kaldkorpus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Anillo_de_divisi%C3%B3n" title="Anillo de división – Spanish" lang="es" hreflang="es" data-title="Anillo de división" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Korpo_(algebro)" title="Korpo (algebro) – Esperanto" lang="eo" hreflang="eo" data-title="Korpo (algebro)" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AD%D9%84%D9%82%D9%87_%D8%AA%D9%82%D8%B3%DB%8C%D9%85" title="حلقه تقسیم – Persian" lang="fa" hreflang="fa" data-title="حلقه تقسیم" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Corps_gauche" title="Corps gauche – French" lang="fr" hreflang="fr" data-title="Corps gauche" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%82%98%EB%88%97%EC%85%88%ED%99%98" title="나눗셈환 – Korean" lang="ko" hreflang="ko" data-title="나눗셈환" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Algebraic structure also called skew field</div> <p>In <a href="/wiki/Algebra" title="Algebra">algebra</a>, a <b>division ring</b>, also called a <b>skew field</b> (or, occasionally, a <b>sfield</b><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>), is a <a href="/wiki/Zero_ring" title="Zero ring">nontrivial</a> <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> in which <a href="/wiki/Division_(mathematics)" title="Division (mathematics)">division</a> by nonzero elements is defined. Specifically, it is a nontrivial ring<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> in which every nonzero element <span class="texhtml mvar" style="font-style:italic;">a</span> has a <a href="/wiki/Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>, that is, an element usually denoted <span class="texhtml"><i>a</i><sup>–1</sup></span>, such that <span class="texhtml"><i>a<span class="nowrap"> </span>a</i><sup>–1</sup> = <i>a</i><sup>–1</sup><span class="nowrap"> </span><i>a</i> = 1</span>. So, (right) <i>division</i> may be defined as <span class="texhtml"><i>a</i> / <i>b</i> = <i>a</i><span class="nowrap"> </span><i>b</i><sup>–1</sup></span>, but this notation is avoided, as one may have <span class="texhtml"><i>a<span class="nowrap"> </span>b</i><sup>–1</sup> ≠ <i>b</i><sup>–1</sup><span class="nowrap"> </span><i>a</i></span>. </p><p>A commutative division ring is a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>. <a href="/wiki/Wedderburn%27s_little_theorem" title="Wedderburn's little theorem">Wedderburn's little theorem</a> asserts that all finite division rings are commutative and therefore <a href="/wiki/Finite_field" title="Finite field">finite fields</a>. </p><p>Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields".<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> In some languages, such as <a href="/wiki/French_language" title="French language">French</a>, the word equivalent to "field" ("corps") is used for both commutative and noncommutative cases, and the distinction between the two cases is made by adding qualificatives such as "corps commutatif" (commutative field) or "corps gauche" (skew field). </p><p>All division rings are <a href="/wiki/Simple_ring" title="Simple ring">simple</a>. 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title="Racks and quandles">Rack and quandle</a></li> <li><a href="/wiki/Quasigroup" title="Quasigroup">Quasigroup and loop</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Abelian_group" title="Abelian group">Abelian group</a></li> <li><a href="/wiki/Magma_(algebra)" title="Magma (algebra)">Magma</a></li> <li><a href="/wiki/Lie_group" title="Lie group">Lie group</a></li></ul> </div> <i><a href="/wiki/Group_theory" title="Group theory">Group theory</a></i></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">Ring</a></li> <li><a href="/wiki/Rng_(algebra)" title="Rng (algebra)">Rng</a></li> <li><a href="/wiki/Semiring" title="Semiring">Semiring</a></li> <li><a href="/wiki/Near-ring" title="Near-ring">Near-ring</a></li> <li><a href="/wiki/Commutative_ring" title="Commutative ring">Commutative ring</a></li> <li><a href="/wiki/Domain_(ring_theory)" title="Domain (ring theory)">Domain</a></li> <li><a href="/wiki/Integral_domain" title="Integral domain">Integral domain</a></li> <li><a href="/wiki/Field_(mathematics)" title="Field (mathematics)">Field</a></li> <li><a class="mw-selflink selflink">Division ring</a></li> <li><a href="/wiki/Lie_algebra#Lie_ring" title="Lie algebra">Lie ring</a></li></ul> </div> <i><a href="/wiki/Ring_theory" title="Ring theory">Ring theory</a></i></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="text-align:center;;color: var(--color-base)"><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a>-like</div><div class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"><div class="hlist"> <ul><li><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice</a></li> <li><a href="/wiki/Semilattice" title="Semilattice">Semilattice</a></li> <li><a href="/wiki/Complemented_lattice" title="Complemented lattice">Complemented lattice</a></li> <li><a href="/wiki/Total_order" title="Total order">Total order</a></li> <li><a href="/wiki/Heyting_algebra" title="Heyting algebra">Heyting algebra</a></li> <li><a href="/wiki/Boolean_algebra_(structure)" title="Boolean algebra (structure)">Boolean algebra</a></li></ul> </div> <ul><li><a href="/wiki/Map_of_lattices" title="Map of lattices">Map of lattices</a></li> <li><i><a href="/wiki/Lattice_(order)" title="Lattice (order)">Lattice theory</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible 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class="sidebar-list-content mw-collapsible-content" style="border-top:1px solid #aaa;border-bottom:1px solid #aaa;"> <ul><li><a href="/wiki/Algebra_over_a_field" title="Algebra over a field">Algebra</a></li></ul> <div class="hlist"> <ul><li><a href="/wiki/Associative_algebra" title="Associative algebra">Associative</a></li> <li><a href="/wiki/Non-associative_algebra" title="Non-associative algebra">Non-associative</a></li> <li><a href="/wiki/Composition_algebra" title="Composition algebra">Composition algebra</a></li> <li><a href="/wiki/Lie_algebra" title="Lie algebra">Lie algebra</a></li> <li><a href="/wiki/Graded_ring" title="Graded ring">Graded</a></li> <li><a href="/wiki/Bialgebra" title="Bialgebra">Bialgebra</a></li> <li><a href="/wiki/Hopf_algebra" title="Hopf algebra">Hopf algebra</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Algebraic_structures" title="Template:Algebraic structures"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Algebraic_structures" title="Template talk:Algebraic structures"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Algebraic_structures" title="Special:EditPage/Template:Algebraic structures"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Relation_to_fields_and_linear_algebra">Relation to fields and linear algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Division_ring&action=edit&section=1" title="Edit section: Relation to fields and linear algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>All fields are division rings, and every non-field division ring is noncommutative. The best known example is the ring of <a href="/wiki/Quaternion" title="Quaternion">quaternions</a>. If one allows only <a href="/wiki/Rational_number" title="Rational number">rational</a> instead of <a href="/wiki/Real_number" title="Real number">real</a> coefficients in the constructions of the quaternions, one obtains another division ring. In general, if <span class="texhtml"><i>R</i></span> is a ring and <span class="texhtml"><i>S</i></span> is a <a href="/wiki/Simple_module" title="Simple module">simple module</a> over <span class="texhtml"><i>R</i></span>, then, by <a href="/wiki/Schur%27s_lemma" title="Schur's lemma">Schur's lemma</a>, the <a href="/wiki/Endomorphism_ring" title="Endomorphism ring">endomorphism ring</a> of <span class="texhtml"><i>S</i></span> is a division ring;<sup id="cite_ref-FOOTNOTELam2001''[httpsbooksgooglecombooksidf15FyZuZ3-4CpgPA33dq22Schur27sLemma22_Schur's_Lemma]'',_p._33,_at_[[Google_Books]]_8-0" class="reference"><a href="#cite_note-FOOTNOTELam2001''[httpsbooksgooglecombooksidf15FyZuZ3-4CpgPA33dq22Schur27sLemma22_Schur's_Lemma]'',_p._33,_at_[[Google_Books]]-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> every division ring arises in this fashion from some simple module. </p><p>Much of <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> may be formulated, and remains correct, for <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">modules</a> over a division ring <span class="texhtml"><i>D</i></span> instead of <a href="/wiki/Vector_space" title="Vector space">vector spaces</a> over a field. Doing so, one must specify whether one is considering right or left modules, and some care is needed in properly distinguishing left and right in formulas. In particular, every module has a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a>, and <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a> can be used. So, everything that can be defined with these tools works on division algebras. <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrices</a> and their products are defined similarly.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (July 2023)">citation needed</span></a></i>]</sup> However, a matrix that is left <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible</a> need not to be right invertible, and if it is, its right inverse can differ from its left inverse. (See <i><a href="/wiki/Generalized_inverse#One-sided_inverse" title="Generalized inverse">Generalized inverse § One-sided inverse</a></i>.) </p><p><a href="/wiki/Determinant" title="Determinant">Determinants</a> are not defined over noncommutative division algebras, and everything that requires this concept cannot be generalized to noncommutative division algebras. </p><p>Working in coordinates, elements of a finite-dimensional right module can be represented by column vectors, which can be multiplied on the right by scalars, and on the left by matrices (representing linear maps); for elements of a finite-dimensional left module, row vectors must be used, which can be multiplied on the left by scalars, and on the right by matrices. The dual of a right module is a left module, and vice versa. The transpose of a matrix must be viewed as a matrix over the opposite division ring <span class="texhtml"><i>D</i><sup>op</sup></span> in order for the rule <span class="texhtml">(<i>AB</i>)<sup>T</sup> = <i>B</i><sup>T</sup><i>A</i><sup>T</sup></span> to remain valid. </p><p>Every module over a division ring is <a href="/wiki/Free_module" title="Free module">free</a>; that is, it has a basis, and all bases of a module <a href="/wiki/Invariant_basis_number" title="Invariant basis number">have the same number of elements</a>. Linear maps between finite-dimensional modules over a division ring can be described by <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>; the fact that linear maps by definition commute with scalar multiplication is most conveniently represented in notation by writing them on the <i>opposite</i> side of vectors as scalars are. The <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a> algorithm remains applicable. The column rank of a matrix is the dimension of the right module generated by the columns, and the row rank is the dimension of the left module generated by the rows; the same proof as for the vector space case can be used to show that these ranks are the same and define the rank of a matrix. </p><p>Division rings are the only <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">rings</a> over which every module is free: a ring <span class="texhtml"><i>R</i></span> is a division ring if and only if every <span class="texhtml"><i>R</i></span>-module is <a href="/wiki/Free_module" title="Free module">free</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Center_of_a_ring" class="mw-redirect" title="Center of a ring">center</a> of a division ring is commutative and therefore a field.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Every division ring is therefore a <a href="/wiki/Division_algebra" title="Division algebra">division algebra</a> over its center. Division rings can be roughly classified according to whether or not they are finite dimensional or infinite dimensional over their centers. The former are called <i>centrally finite</i> and the latter <i>centrally infinite</i>. Every field is one dimensional over its center. The ring of <a href="/wiki/Hamiltonian_quaternions" class="mw-redirect" title="Hamiltonian quaternions">Hamiltonian quaternions</a> forms a four-dimensional algebra over its center, which is isomorphic to the real numbers. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Division_ring&action=edit&section=2" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>As noted above, all <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a> are division rings.</li> <li>The <a href="/wiki/Quaternion" title="Quaternion">quaternions</a> form a noncommutative division ring.</li> <li>The subset of the quaternions <span class="texhtml"><i>a</i> + <i>bi</i> + <i>cj</i> + <i>dk</i></span>, such that <span class="texhtml mvar" style="font-style:italic;">a</span>, <span class="texhtml mvar" style="font-style:italic;">b</span>, <span class="texhtml mvar" style="font-style:italic;">c</span>, and <span class="texhtml mvar" style="font-style:italic;">d</span> belong to a fixed subfield of the <a href="/wiki/Real_number" title="Real number">real numbers</a>, is a noncommutative division ring. When this subfield is the field of <a href="/wiki/Rational_number" title="Rational number">rational numbers</a>, this is the division ring of <i>rational quaternions</i>.</li> <li>Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma :\mathbb {C} \to \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma :\mathbb {C} \to \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36b05874c6cc3ed86307231fb488a97420fcb631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.237ex; height:2.176ex;" alt="{\displaystyle \sigma :\mathbb {C} \to \mathbb {C} }"></span> be an <a href="/wiki/Automorphism" title="Automorphism">automorphism</a> of the field <span class="nowrap"><span class="mwe-math-element"><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 {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7" 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 {C} }"></span>.</span> Let <span class="mwe-math-element"><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 {C} ((z,\sigma ))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>z</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ((z,\sigma ))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/652586d8eb9e5c3cbd9dd5c9a8c435d36dadcce2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.748ex; height:2.843ex;" alt="{\displaystyle \mathbb {C} ((z,\sigma ))}"></span> denote the ring of <a href="/wiki/Formal_Laurent_series" class="mw-redirect" title="Formal Laurent series">formal Laurent series</a> with complex coefficients, wherein multiplication is defined as follows: instead of simply allowing coefficients to commute directly with the indeterminate <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}"></span>,</span> for <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in \mathbb {C} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c2956c000d6b14cfab5918e7129edd52af31706" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle \alpha \in \mathbb {C} }"></span>,</span> define <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z^{i}\alpha :=\sigma ^{i}(\alpha )z^{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mi>α</mi> <mo>:=</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z^{i}\alpha :=\sigma ^{i}(\alpha )z^{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/580f65467ab31fc929cee6c832380fac0604dfcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.44ex; height:3.176ex;" alt="{\displaystyle z^{i}\alpha :=\sigma ^{i}(\alpha )z^{i}}"></span> for each index <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae4e6baa4f5f76345756ffed7375ad64b0f0ecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.193ex; height:2.176ex;" alt="{\displaystyle i\in \mathbb {Z} }"></span>.</span> If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span> is a non-trivial automorphism of <a href="/wiki/Complex_number" title="Complex number">complex numbers</a> (such as <a href="/wiki/Complex_conjugate" title="Complex conjugate">the conjugation</a>), then the resulting ring of Laurent series is a noncommutative division ring known as a <i>skew Laurent series ring</i>;<sup id="cite_ref-FOOTNOTELam200110_11-0" class="reference"><a href="#cite_note-FOOTNOTELam200110-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> if <span class="texhtml"><i>σ</i> = <a href="/wiki/Identity_function" title="Identity function">id</a></span> then it features the <a href="/wiki/Ring_of_formal_Laurent_series" class="mw-redirect" title="Ring of formal Laurent series">standard multiplication of formal series</a>. This concept can be generalized to the ring of Laurent series over any fixed field <span class="nowrap"><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" 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}"></span>,</span> given a nontrivial <span class="nowrap"><span class="mwe-math-element"><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="https://wikimedia.org/api/rest_v1/media/math/render/svg/545fd099af8541605f7ee55f08225526be88ce57" 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}"></span>-automorphism</span> <span class="nowrap"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }"></span>.</span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Main_theorems">Main theorems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Division_ring&action=edit&section=3" title="Edit section: Main theorems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b><a href="/wiki/Wedderburn%27s_little_theorem" title="Wedderburn's little theorem">Wedderburn's little theorem</a></b>: All finite division rings are commutative and therefore <a href="/wiki/Finite_field" title="Finite field">finite fields</a>. (<a href="/wiki/Ernst_Witt" title="Ernst Witt">Ernst Witt</a> gave a simple proof.) </p><p><b><a href="/wiki/Frobenius_theorem_(real_division_algebras)" title="Frobenius theorem (real division algebras)">Frobenius theorem</a></b>: The only finite-dimensional associative division algebras over the reals are the reals themselves, the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, and the <a href="/wiki/Quaternion" title="Quaternion">quaternions</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Related_notions">Related notions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Division_ring&action=edit&section=4" title="Edit section: Related notions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Division rings <i>used to be</i> called "fields" in an older usage. In many languages, a word meaning "body" is used for division rings, in some languages designating either commutative or noncommutative division rings, while in others specifically designating commutative division rings (what we now call fields in English). A more complete comparison is found in the article on <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">fields</a>. </p><p>The name "skew field" has an interesting <a href="/wiki/Lexical_semantics" title="Lexical semantics">semantic</a> feature: a modifier (here "skew") <i>widens</i> the scope of the base term (here "field"). Thus a field is a particular type of skew field, and not all skew fields are fields. </p><p>While division rings and algebras as discussed here are assumed to have associative multiplication, <a href="/wiki/Division_algebra#Not_necessarily_associative_division_algebras" title="Division algebra">nonassociative division algebras</a> such as the <a href="/wiki/Octonion" title="Octonion">octonions</a> are also of interest. </p><p>A <a href="/wiki/Near-field_(mathematics)" title="Near-field (mathematics)">near-field</a> is an algebraic structure similar to a division ring, except that it has only one of the two <a href="/wiki/Distributive_law" class="mw-redirect" title="Distributive law">distributive laws</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Division_ring&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Hua%27s_identity" title="Hua's identity">Hua's identity</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Division_ring&action=edit&section=6" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://proofwiki.org/wiki/Definition:Skew_Field">"Definition:Skew Field - ProofWiki"</a>. <i>proofwiki.org</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-10-13</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=proofwiki.org&rft.atitle=Definition%3ASkew+Field+-+ProofWiki&rft_id=https%3A%2F%2Fproofwiki.org%2Fwiki%2FDefinition%3ASkew_Field&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADivision+ring" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHua1949" class="citation journal cs1">Hua, Loo-Keng (1949). <a rel="nofollow" class="external text" href="https://pnas.org/doi/full/10.1073/pnas.35.9.533">"Some Properties of a Sfield"</a>. <i>Proceedings of the National Academy of Sciences</i>. <b>35</b> (9): 533–537. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.35.9.533">10.1073/pnas.35.9.533</a>. <a href="/wiki/ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0027-8424">0027-8424</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1063075">1063075</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/16588934">16588934</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+National+Academy+of+Sciences&rft.atitle=Some+Properties+of+a+Sfield&rft.volume=35&rft.issue=9&rft.pages=533-537&rft.date=1949&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC1063075%23id-name%3DPMC&rft.issn=0027-8424&rft_id=info%3Apmid%2F16588934&rft_id=info%3Adoi%2F10.1073%2Fpnas.35.9.533&rft.aulast=Hua&rft.aufirst=Loo-Keng&rft_id=https%3A%2F%2Fpnas.org%2Fdoi%2Ffull%2F10.1073%2Fpnas.35.9.533&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADivision+ring" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">In this article, rings have a <span class="texhtml">1</span>.</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">1948, Rings and Ideals. Northampton, Mass., Mathematical Association of America</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFArtin1965" class="citation cs2">Artin, Emil (1965), Serge Lang; John T. Tate (eds.), <i>Collected Papers</i>, New York: Springer</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Collected+Papers&rft.place=New+York&rft.pub=Springer&rft.date=1965&rft.aulast=Artin&rft.aufirst=Emil&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADivision+ring" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrauer1932" class="citation cs2">Brauer, Richard (1932), "Über die algebraische Struktur von Schiefkörpern", <i>Journal für die reine und angewandte Mathematik</i> (166.4): 103–252</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+f%C3%BCr+die+reine+und+angewandte+Mathematik&rft.atitle=%C3%9Cber+die+algebraische+Struktur+von+Schiefk%C3%B6rpern&rft.issue=166.4&rft.pages=103-252&rft.date=1932&rft.aulast=Brauer&rft.aufirst=Richard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADivision+ring" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Within the English language area the terms "skew field" and "sfield" were mentioned 1948 by Neal McCoy<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> as "sometimes used in the literature", and since 1965 <i>skewfield</i> has an entry in the <a href="/wiki/OED" class="mw-redirect" title="OED">OED</a>. The German term <a href="/w/index.php?title=Schiefk%C3%B6rper&action=edit&redlink=1" class="new" title="Schiefkörper (page does not exist)">Schiefkörper</a><sup class="noprint" style="font-style: normal;"> [<a href="https://de.wikipedia.org/wiki/Schiefk%C3%B6rper" class="extiw" title="de:Schiefkörper">de</a>]</sup> is documented, as a suggestion by <a href="/wiki/Van_der_Waerden" class="mw-redirect" title="Van der Waerden">van der Waerden</a>, in a 1927 text by <a href="/wiki/Emil_Artin" title="Emil Artin">Emil Artin</a>,<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> and was used by <a href="/wiki/Emmy_Noether" title="Emmy Noether">Emmy Noether</a> as lecture title in 1928.<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></span> </li> <li id="cite_note-FOOTNOTELam2001''[httpsbooksgooglecombooksidf15FyZuZ3-4CpgPA33dq22Schur27sLemma22_Schur's_Lemma]'',_p._33,_at_[[Google_Books]]-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELam2001''[httpsbooksgooglecombooksidf15FyZuZ3-4CpgPA33dq22Schur27sLemma22_Schur's_Lemma]'',_p._33,_at_[[Google_Books]]_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLam2001">Lam (2001)</a>, <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=f15FyZuZ3-4C&pg=PA33&dq=%22Schur%27s+Lemma%22">Schur's Lemma</a></i>, p. 33, at <a href="/wiki/Google_Books" title="Google Books">Google Books</a></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Grillet, Pierre Antoine. Abstract algebra. Vol. 242. Springer Science & Business Media, 2007</span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Simple commutative rings are fields. See <a href="#CITEREFLam2001">Lam (2001)</a>, <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=f15FyZuZ3-4C&pg=PA39&dq=%22simple+commutative+rings%22">simple commutative rings</a></i>, p. 39, at <a href="/wiki/Google_Books" title="Google Books">Google Books</a> and <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=f15FyZuZ3-4C&pg=PA45&dq=%22center+of+a+simple+ring%22">exercise 3.4</a></i>, p. 45, at <a href="/wiki/Google_Books" title="Google Books">Google Books</a></span> </li> <li id="cite_note-FOOTNOTELam200110-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTELam200110_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFLam2001">Lam (2001)</a>, p. 10.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Division_ring&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLam2001" class="citation book cs1"><a href="/wiki/Tsit_Yuen_Lam" title="Tsit Yuen Lam">Lam, Tsit-Yuen</a> (2001). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=f15FyZuZ3-4C&q=%22division+ring%22"><i>A first course in noncommutative rings</i></a>. <a href="/wiki/Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>. Vol. 131 (2nd ed.). Springer. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95183-0" title="Special:BookSources/0-387-95183-0"><bdi>0-387-95183-0</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0980.16001">0980.16001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+first+course+in+noncommutative+rings&rft.series=Graduate+Texts+in+Mathematics&rft.edition=2nd&rft.pub=Springer&rft.date=2001&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0980.16001%23id-name%3DZbl&rft.isbn=0-387-95183-0&rft.aulast=Lam&rft.aufirst=Tsit-Yuen&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Df15FyZuZ3-4C%26q%3D%2522division%2Bring%2522&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADivision+ring" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Division_ring&action=edit&section=8" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCohn1995" class="citation book cs1"><a href="/wiki/Paul_Cohn" title="Paul Cohn">Cohn, P.M.</a> (1995). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/skewfieldstheory0000cohn"><i>Skew fields. Theory of general division rings</i></a></span>. Encyclopedia of Mathematics and Its Applications. Vol. 57. Cambridge: <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-43217-0" title="Special:BookSources/0-521-43217-0"><bdi>0-521-43217-0</bdi></a>. <a href="/wiki/Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0840.16001">0840.16001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Skew+fields.+Theory+of+general+division+rings&rft.place=Cambridge&rft.series=Encyclopedia+of+Mathematics+and+Its+Applications&rft.pub=Cambridge+University+Press&rft.date=1995&rft_id=https%3A%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0840.16001%23id-name%3DZbl&rft.isbn=0-521-43217-0&rft.aulast=Cohn&rft.aufirst=P.M.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fskewfieldstheory0000cohn&rfr_id=info%3Asid%2Fen.wikipedia.org%3ADivision+ring" class="Z3988"></span></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Division_ring&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><a rel="nofollow" class="external text" href="https://planetmath.org/?op=getobj&from=objects&id=3627">Proof of Wedderburn's Theorem at Planet Math</a></li> <li><a rel="nofollow" class="external text" href="https://math.stackexchange.com/q/75866">Grillet's Abstract Algebra, section VIII.5's characterization of division rings via their free modules. </a></li></ul> </div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Division_ring&oldid=1250953832">https://en.wikipedia.org/w/index.php?title=Division_ring&oldid=1250953832</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Category</a>: <ul><li><a href="/wiki/Category:Ring_theory" title="Category:Ring theory">Ring theory</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:All_articles_with_unsourced_statements" title="Category:All articles with unsourced statements">All articles with unsourced statements</a></li><li><a href="/wiki/Category:Articles_with_unsourced_statements_from_July_2023" title="Category:Articles with unsourced statements from July 2023">Articles with unsourced statements from July 2023</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 13 October 2024, at 14:46<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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