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<title>Noncommutative ring</title>

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</style><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist" style="width: 20.5em;"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.4em;"><span style="font-size: 8pt; font-weight: none"><a href="Algebraic_structure" title="Algebraic structure">Algebraic structure</a> → Ring theory</span><br><a href="Ring_theory" title="Ring theory">Ring theory</a></th></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)">Basic concepts</div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><b><a href="Ring_(mathematics)" title="Ring (mathematics)">Rings</a></b>

<dl><dd>• <a href="Subring" title="Subring">Subrings</a></dd>

<dd>• <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">Ideal</a></dd>

<dd>• <a href="Quotient_ring" title="Quotient ring">Quotient ring</a>

<dl><dd>• <a href="Fractional_ideal" title="Fractional ideal">Fractional ideal</a></dd>

<dd>• <a href="Total_ring_of_fractions" title="Total ring of fractions">Total ring of fractions</a></dd></dl></dd>

<dd>• <a href="Product_of_rings" title="Product of rings">Product of rings</a></dd>

<dd>• <a href="Free_product_of_associative_algebras" title="Free product of associative algebras">Free product of associative algebras</a></dd>

<dd>• <a href="Tensor_product_of_algebras" title="Tensor product of algebras">Tensor product of algebras</a></dd></dl>

<p><b><a href="Ring_homomorphism" title="Ring homomorphism">Ring homomorphisms</a></b>

</p>

<dl><dd>• <a href="Kernel_(algebra)#Ring_homomorphisms" title="Kernel (algebra)">Kernel</a></dd>

<dd>• <a href="Inner_automorphism#Ring_case" title="Inner automorphism">Inner automorphism</a></dd>

<dd>• <a href="Frobenius_endomorphism" title="Frobenius endomorphism">Frobenius endomorphism</a></dd></dl>

<p><b><a href="Algebraic_structure" title="Algebraic structure">Algebraic structures</a></b>

</p>

<dl><dd>• <a href="Module_(mathematics)" title="Module (mathematics)">Module</a></dd>

<dd>• <a href="Associative_algebra" title="Associative algebra">Associative algebra</a></dd>

<dd>• <a href="Graded_ring" title="Graded ring">Graded ring</a></dd>

<dd>• <a href="Involutive_ring" class="mw-redirect" title="Involutive ring">Involutive ring</a></dd>

<dd>• <a href="Category_of_rings" title="Category of rings">Category of rings</a>

<dl><dd>• <a href="Integer" title="Integer">Initial 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 \mathbb {Z} }">

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</math></span><img src="./449494a083e0a1fda2b61c62b2f09b6bee4633dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }" loading="lazy"></span></dd>

<dd>• <a href="Zero_ring" title="Zero ring">Terminal 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 0=\mathbb {Z} /1\mathbb {Z} }">

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<annotation encoding="application/x-tex">{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }</annotation>

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</math></span><img src="./e45ab495cb8cfbac68a9322af662c3d6c7dbe494.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.686ex; height:2.843ex;" alt="{\displaystyle 0=\mathbb {Z} /1\mathbb {Z} }" loading="lazy"></span></dd></dl></dd></dl>

<p><b>Related structures</b>

</p>

<dl><dd>• <a href="Field_(mathematics)" title="Field (mathematics)">Field</a>

<dl><dd>• <a href="Finite_field" title="Finite field">Finite field</a></dd></dl></dd>

<dd>• <a href="Non-associative_ring" class="mw-redirect" title="Non-associative ring">Non-associative ring</a>

<dd>• <a href="Jordan_ring" class="mw-redirect" title="Jordan ring">Jordan ring</a></dd></dl></dd>

<dd>• <a href="Semiring" title="Semiring">Semiring</a>

<dl><dd>• <a href="Semifield" title="Semifield">Semifield</a></dd></dl></dd></dl></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="Commutative_algebra" title="Commutative algebra">Commutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><b><a href="Commutative_ring" title="Commutative ring">Commutative rings</a></b>

<dl><dd>• <a href="Integral_domain" title="Integral domain">Integral domain</a>

<dl><dd>• <a href="Integrally_closed_domain" title="Integrally closed domain">Integrally closed domain</a></dd>

<dd>• <a href="GCD_domain" title="GCD domain">GCD domain</a></dd>

<dd>• <a href="Unique_factorization_domain" title="Unique factorization domain">Unique factorization domain</a></dd>

<dd>• <a href="Principal_ideal_domain" title="Principal ideal domain">Principal ideal domain</a></dd>

<dd>• <a href="Euclidean_domain" title="Euclidean domain">Euclidean domain</a></dd>

<dd>• <a href="Field_(mathematics)" title="Field (mathematics)">Field</a>

<dl><dd>• <a href="Finite_field" title="Finite field">Finite field</a></dd></dl></dd>

<dd>• <a href="Polynomial_ring" title="Polynomial ring">Polynomial ring</a></dd>

<dd>• <a href="Formal_power_series_ring" class="mw-redirect" title="Formal power series ring">Formal power series ring</a></dd></dl></dd></dl>

<p><b><a href="Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></b>

</p>

<dl><dd>• <a href="Algebraic_number_field" title="Algebraic number field">Algebraic number field</a></dd>

<dd>• <a href="Integers_modulo_n" class="mw-redirect" title="Integers modulo n">Integers modulo <span class="texhtml mvar" style="font-style:italic;">n</span></a></dd>

<dd>• <a href="Ring_of_integers" title="Ring of integers">Ring of integers</a></dd>

<dd>• <a href="P-adic_integer" class="mw-redirect" title="P-adic integer"><i>p</i>-adic integers</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} _{p}}">

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

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</math></span><img src="./dbc1df7227ef11fe88dccd2dae3adc7bbdeae5f4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.609ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} _{p}}" loading="lazy"></span></dd>

<dd>• <a href="P-adic_number" title="P-adic number"><i>p</i>-adic numbers</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Q} _{p}}">

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</math></span><img src="./35f44bc6894c682710705f3ea74f33042e0acc3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.867ex; height:2.843ex;" alt="{\displaystyle \mathbb {Q} _{p}}" loading="lazy"></span></dd>

<dd>• <a href="Pr%C3%BCfer_group#The_Prüfer_group_as_a_ring" title="Prüfer group">Prüfer <i>p</i>-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 \mathbb {Z} (p^{\infty })}">

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</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="Noncommutative_algebra" class="mw-redirect" title="Noncommutative algebra">Noncommutative algebra</a></div><div class="sidebar-list-content mw-collapsible-content" style="text-align: left;"><b></b>

<dl><dd>• <a href="Division_ring" title="Division ring">Division ring</a></dd>

<dd>• <a href="Semiprimitive_ring" title="Semiprimitive ring">Semiprimitive ring</a></dd>

<dd>• <a href="Simple_ring" title="Simple ring">Simple ring</a></dd>

<dd>• <a href="Commutator_(ring_theory)" class="mw-redirect" title="Commutator (ring theory)">Commutator</a></dd></dl>

<p><b><a href="Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a></b>

</p><p><b><a href="Free_algebra" title="Free algebra">Free algebra</a></b>

</p><p><b><a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a></b>

</p>

<dl><dd>• <a href="Geometric_algebra" title="Geometric algebra">Geometric algebra</a></dd></dl>

<b><a href="Operator_algebra" title="Operator algebra">Operator algebra</a></b></div></div></td>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>noncommutative ring</b> is a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> whose multiplication is not <a href="Commutative_property" title="Commutative property">commutative</a>; that is, there exist <i>a</i> and <i>b</i> in the ring such that <i>ab</i> and <i>ba</i> are different. Equivalently, a <i>noncommutative ring</i> is a ring that is not a <a href="Commutative_ring" title="Commutative ring">commutative ring</a>.

</p><p><b>Noncommutative algebra</b> is the part of <a href="Ring_theory" title="Ring theory">ring theory</a> devoted to study of properties of the noncommutative rings, including the properties that apply also to commutative rings.

</p><p>Sometimes the term <i>noncommutative ring</i> is used instead of <i>ring</i> to refer to an unspecified ring which is not necessarily commutative, and hence may be commutative. Generally, this is for emphasizing that the studied properties are not restricted to commutative rings, as, in many contexts, <i>ring</i> is used as a shorthand for <i>commutative ring</i>.

</p><p>Although some authors do not assume that rings have a multiplicative identity, in this article we make that assumption unless stated otherwise.

</p>

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

<p>Some examples of noncommutative rings:

</p>

<ul><li>The <a href="Matrix_ring" title="Matrix ring">matrix ring</a> of <i>n</i>-by-<i>n</i> matrices over the <a href="Real_number" title="Real number">real numbers</a>, where <span class="nowrap"><i>n</i> > 1</span></li>

<li>Hamilton's <a href="Quaternion" title="Quaternion">quaternions</a></li>

<li>Any <a href="Group_ring" title="Group ring">group ring</a> constructed from a group that is not <a href="Abelian_group" title="Abelian group">abelian</a></li></ul>

<p>Some examples of rings that are not typically commutative (but may be commutative in simple cases):

</p>

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} \langle x_{1},\ldots ,x_{n}\rangle }</annotation>

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</math></span><img src="./654f4958a09270702fda3809520432ba773d4766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.47ex; height:2.843ex;" alt="{\displaystyle \mathbb {Z} \langle x_{1},\ldots ,x_{n}\rangle }" loading="lazy"></span> generated by a finite set, an example of two non-equal elements being <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 2x_{1}x_{2}+x_{2}x_{1}\neq 3x_{1}x_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle 2x_{1}x_{2}+x_{2}x_{1}\neq 3x_{1}x_{2}}</annotation>

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</math></span><img src="./f943dc44b729d4414329d68e516aff3968e7fe16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.567ex; height:2.676ex;" alt="{\displaystyle 2x_{1}x_{2}+x_{2}x_{1}\neq 3x_{1}x_{2}}" loading="lazy"></span></li>

<li>The <a href="Weyl_algebra" title="Weyl algebra">Weyl algebra</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 A_{n}(\mathbb {C} )}">

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

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<annotation encoding="application/x-tex">{\displaystyle A_{1}(\mathbb {C} )\cong \mathbb {C} \langle x,y\rangle /(xy-yx-1)}</annotation>

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</math></span><img src="./20efbba66492a8cf3cad3a9100e0bbbf9291db83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.175ex; height:2.843ex;" alt="{\displaystyle A_{1}(\mathbb {C} )\cong \mathbb {C} \langle x,y\rangle /(xy-yx-1)}" loading="lazy"></span>, where the ideal corresponds to the <a href="Commutator#Ring_theory" title="Commutator">commutator</a></li>

<li>The quotient ring <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} \langle x_{1},\ldots ,x_{n}\rangle /(x_{i}x_{j}-q_{ij}x_{j}x_{i})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} \langle x_{1},\ldots ,x_{n}\rangle /(x_{i}x_{j}-q_{ij}x_{j}x_{i})}</annotation>

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</math></span><img src="./9d0cac236ab37bf105f2aea56b00b761a3b9005c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.661ex; height:3.009ex;" alt="{\displaystyle \mathbb {C} \langle x_{1},\ldots ,x_{n}\rangle /(x_{i}x_{j}-q_{ij}x_{j}x_{i})}" loading="lazy"></span>, called a quantum plane, where <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 q_{ij}\in \mathbb {C} }">

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<annotation encoding="application/x-tex">{\displaystyle q_{ij}\in \mathbb {C} }</annotation>

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</math></span><img src="./853fac5791816329c5b6ecd60dec0b605cd3a4ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.033ex; height:2.843ex;" alt="{\displaystyle q_{ij}\in \mathbb {C} }" loading="lazy"></span></li>

<li>Any <a href="Clifford_algebra" title="Clifford algebra">Clifford algebra</a> can be described explicitly using an algebra presentation: given an <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} }">

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

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</math></span><img src="./573f72afae7df709959ab1a58cd643743466a187.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.176ex;" alt="{\displaystyle \mathbb {F} }" loading="lazy"></span>-vector space <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}">

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

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</math></span><img src="./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> of dimension <span class="texhtml mvar" style="font-style:italic;">n</span> with a quadratic form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q:V\otimes V\to \mathbb {F} }">

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<annotation encoding="application/x-tex">{\displaystyle q:V\otimes V\to \mathbb {F} }</annotation>

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</math></span><img src="./98f1e71326d01b52ecbd3871429450b37e3f3bf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.456ex; height:2.509ex;" alt="{\displaystyle q:V\otimes V\to \mathbb {F} }" loading="lazy"></span>, the associated Clifford algebra has the presentation <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {F} \langle e_{1},\ldots ,e_{n}\rangle /(e_{i}e_{j}+e_{j}e_{i}-q(e_{i},e_{j}))}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {F} \langle e_{1},\ldots ,e_{n}\rangle /(e_{i}e_{j}+e_{j}e_{i}-q(e_{i},e_{j}))}</annotation>

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</math></span><img src="./e092eb59a5e31a855336462817a81a86fc974337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:37.042ex; height:3.009ex;" alt="{\displaystyle \mathbb {F} \langle e_{1},\ldots ,e_{n}\rangle /(e_{i}e_{j}+e_{j}e_{i}-q(e_{i},e_{j}))}" loading="lazy"></span> for any basis <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{1},\ldots ,e_{n}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{1},\ldots ,e_{n}}</annotation>

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</math></span><img src="./c60c38b7e2450d62e9dc496b89f8e5c96c77cecf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.618ex; height:2.009ex;" alt="{\displaystyle e_{1},\ldots ,e_{n}}" loading="lazy"></span> of <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}">

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

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<li><a href="Superalgebra" title="Superalgebra">Superalgebras</a> are another example of noncommutative rings; they can be presented as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} [x_{1},\ldots ,x_{n}]\langle \theta _{1},\ldots ,\theta _{m}\rangle /(\theta _{i}\theta _{j}+\theta _{j}\theta _{i})}">

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<annotation encoding="application/x-tex">{\displaystyle \mathbb {C} [x_{1},\ldots ,x_{n}]\langle \theta _{1},\ldots ,\theta _{m}\rangle /(\theta _{i}\theta _{j}+\theta _{j}\theta _{i})}</annotation>

</semantics>

</math></span><img src="./113e16379b13e010e4533d0c7172aed8c26d3710.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.572ex; height:3.009ex;" alt="{\displaystyle \mathbb {C} [x_{1},\ldots ,x_{n}]\langle \theta _{1},\ldots ,\theta _{m}\rangle /(\theta _{i}\theta _{j}+\theta _{j}\theta _{i})}" loading="lazy"></span></li>

<li>There are finite noncommutative rings: for example, the <span class="texhtml"><i>n</i></span>-by-<span class="texhtml"><i>n</i></span> matrices over a <a href="Finite_field" title="Finite field">finite field</a>, for <span class="texhtml"><i>n</i> > 1</span>. The smallest noncommutative ring is the ring of the <a href="Upper_triangular_matrices" class="mw-redirect" title="Upper triangular matrices">upper triangular matrices</a> over the field with two elements; it has eight elements and all noncommutative rings with eight elements are <a href="Isomorphic" class="mw-redirect" title="Isomorphic">isomorphic</a> to it or to its <a href="Opposite_ring" title="Opposite ring">opposite</a>.<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></li></ul>

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

<p>Beginning with <a href="Division_ring" title="Division ring">division rings</a> arising from geometry, the study of noncommutative rings has grown into a major area of modern algebra. The theory and exposition of noncommutative rings was expanded and refined in the 19th and 20th centuries by numerous authors. An incomplete list of such contributors includes <a href="Emil_Artin" title="Emil Artin">E. Artin</a>, <a href="Richard_Brauer" title="Richard Brauer">Richard Brauer</a>, <a href="P._M._Cohn" class="mw-redirect" title="P. M. Cohn">P. M. Cohn</a>, <a href="William_Rowan_Hamilton" title="William Rowan Hamilton">W. R. Hamilton</a>, <a href="I._N._Herstein" class="mw-redirect" title="I. N. Herstein">I. N. Herstein</a>, <a href="Nathan_Jacobson" title="Nathan Jacobson">N. Jacobson</a>, <a href="Kiiti_Morita" title="Kiiti Morita">K. Morita</a>, <a href="Emmy_Noether" title="Emmy Noether">E. Noether</a>, <a href="%C3%98ystein_Ore" title="Øystein Ore">Ø. Ore</a>, <a href="Joseph_Wedderburn" title="Joseph Wedderburn">J. Wedderburn</a> and others.

</p>

<div class="mw-heading mw-heading2"><h2 id="Differences_between_commutative_and_noncommutative_algebra">Differences between commutative and noncommutative algebra</h2></div>

<p>Because noncommutative rings of scientific interest are more complicated than commutative rings, their structure, properties and behavior are less well understood. A great deal of work has been done successfully generalizing some results from commutative rings to noncommutative rings. A major difference between rings which are and are not commutative is the necessity to separately consider <a href="Right_ideal" class="mw-redirect" title="Right ideal">right ideals and left ideals</a>. It is common for noncommutative ring theorists to enforce a condition on one of these types of ideals while not requiring it to hold for the opposite side. For commutative rings, the left–right distinction does not exist.

</p>

<div class="mw-heading mw-heading2"><h2 id="Important_classes">Important classes</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Division_rings">Division rings</h3></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Division_ring" title="Division ring">Division ring</a></div>

<p>A division ring, also called a skew field, is a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> in which <a href="Division_(mathematics)" title="Division (mathematics)">division</a> is possible. Specifically, it is a <a href="Zero_ring" title="Zero ring">nonzero</a> ring<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> in which every nonzero element <i>a</i> has a <a href="Multiplicative_inverse" title="Multiplicative inverse">multiplicative inverse</a>, i.e., an element <i>x</i> with <span class="nowrap"><i>a</i> <b>·</b> <i>x</i> = <i>x</i> <b>·</b> <i>a</i> = 1</span>. Stated differently, a ring is a division ring if and only if its <a href="Group_of_units" class="mw-redirect" title="Group of units">group of units</a> is the set of all nonzero elements.

</p><p>Division rings differ from <a href="Field_(mathematics)" title="Field (mathematics)">fields</a> only in that their multiplication is not required to be <a href="Commutative" class="mw-redirect" title="Commutative">commutative</a>. However, by <a href="Wedderburn's_little_theorem" title="Wedderburn's little theorem">Wedderburn's little theorem</a> all finite division rings are commutative and therefore <a href="Finite_field" title="Finite field">finite fields</a>. Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields".

</p>

<div class="mw-heading mw-heading3"><h3 id="Semisimple_rings">Semisimple rings</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Semisimple_ring" class="mw-redirect" title="Semisimple ring">Semisimple ring</a></div>

<p>A <a href="Module_(mathematics)" title="Module (mathematics)">module</a> over a (not necessarily commutative) ring with unity is said to be semisimple (or completely reducible) if it is the <a href="Direct_sum_of_modules" title="Direct sum of modules">direct sum</a> of <a href="Simple_module" title="Simple module">simple</a> (irreducible) submodules.

</p><p>A ring is said to be (left)-semisimple if it is semisimple as a left module over itself. Surprisingly, a left-semisimple ring is also right-semisimple and vice versa. The left/right distinction is therefore unnecessary.

</p>

<div class="mw-heading mw-heading3"><h3 id="Semiprimitive_rings">Semiprimitive rings</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Semiprimitive_ring" title="Semiprimitive ring">Semiprimitive ring</a></div>

<p>A semiprimitive ring or Jacobson semisimple ring or J-semisimple ring is a ring whose <a href="Jacobson_radical" title="Jacobson radical">Jacobson radical</a> is zero. This is a type of ring more general than a <a href="Semisimple_ring" class="mw-redirect" title="Semisimple ring">semisimple ring</a>, but where <a href="Simple_module" title="Simple module">simple modules</a> still provide enough information about the ring. Rings such as the ring of integers are semiprimitive, and an <a href="Artinian_ring" title="Artinian ring">artinian</a> semiprimitive ring is just a <a href="Semisimple_ring" class="mw-redirect" title="Semisimple ring">semisimple ring</a>. Semiprimitive rings can be understood as <a href="Subdirect_product" title="Subdirect product">subdirect products</a> of <a href="Primitive_ring" title="Primitive ring">primitive rings</a>, which are described by the <a href="Jacobson_density_theorem" title="Jacobson density theorem">Jacobson density theorem</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Simple_rings">Simple rings</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Simple_ring" title="Simple ring">Simple ring</a></div>

<p>A simple ring is a non-zero <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> that has no two-sided <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> besides the <a href="Zero_ideal" class="mw-redirect" title="Zero ideal">zero ideal</a> and itself. A simple ring can always be considered as a <a href="Simple_algebra" class="mw-redirect" title="Simple algebra">simple algebra</a>. Rings which are simple as rings but not as <a href="Module_(mathematics)" title="Module (mathematics)">modules</a> do exist: the full <a href="Matrix_ring" title="Matrix ring">matrix ring</a> over a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> does not have any nontrivial ideals (since any ideal of M(<i>n</i>,<i>R</i>) is of the form M(<i>n</i>,<i>I</i>) with <i>I</i> an ideal of <i>R</i>), but has nontrivial left ideals (namely, the sets of matrices which have some fixed zero columns).

</p><p>According to the <a href="Artin%E2%80%93Wedderburn_theorem" class="mw-redirect" title="Artin–Wedderburn theorem">Artin–Wedderburn theorem</a>, every simple ring that is left or right <a href="Artinian_ring" title="Artinian ring">Artinian</a> is a <a href="Matrix_ring" title="Matrix ring">matrix ring</a> over a <a href="Division_ring" title="Division ring">division ring</a>. In particular, the only simple rings that are a finite-dimensional <a href="Vector_space" title="Vector space">vector space</a> over the <a href="Real_number" title="Real number">real numbers</a> are rings of matrices over either the real numbers, the <a href="Complex_number" title="Complex number">complex numbers</a>, or the <a href="Quaternion" title="Quaternion">quaternions</a>.

</p><p>Any quotient of a ring by a <a href="Maximal_ideal" title="Maximal ideal">maximal ideal</a> is a simple ring. In particular, a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> is a simple ring. A ring <i>R</i> is simple if and only if its <a href="Opposite_ring" title="Opposite ring">opposite ring</a> <i>R</i><sup>o</sup> is simple.

</p><p>An example of a simple ring that is not a matrix ring over a division ring is the <a href="Weyl_algebra" title="Weyl algebra">Weyl algebra</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Important_theorems">Important theorems</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Wedderburn's_little_theorem">Wedderburn's little theorem</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Wedderburn's_little_theorem" title="Wedderburn's little theorem">Wedderburn's little theorem</a></div>

<p>Wedderburn's little theorem states that every <a href="Finite_set" title="Finite set">finite</a> <a href="Domain_(ring_theory)" title="Domain (ring theory)">domain</a> is a <a href="Field_(mathematics)" title="Field (mathematics)">field</a>. In other words, for <a href="Finite_ring" title="Finite ring">finite rings</a>, there is no distinction between domains, <a href="Division_ring" title="Division ring">division rings</a> and fields.

</p><p>The <a href="Artin%E2%80%93Zorn_theorem" title="Artin–Zorn theorem">Artin–Zorn theorem</a> generalizes the theorem to <a href="Alternative_ring" class="mw-redirect" title="Alternative ring">alternative rings</a>: every finite simple alternative ring is a field.<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>

</p>

<div class="mw-heading mw-heading3"><h3 id="Artin–Wedderburn_theorem">Artin–Wedderburn theorem</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Artin%E2%80%93Wedderburn_theorem" class="mw-redirect" title="Artin–Wedderburn theorem">Artin–Wedderburn theorem</a></div>

<p>The Artin–Wedderburn theorem is a <a href="Classification_theorem" title="Classification theorem">classification theorem</a> for <a href="Semisimple_ring" class="mw-redirect" title="Semisimple ring">semisimple rings</a> and <a href="Semisimple_algebra" title="Semisimple algebra">semisimple algebras</a>. The theorem states that an (Artinian)<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> semisimple ring <i>R</i> is isomorphic to a <a href="Product_of_rings" title="Product of rings">product</a> of finitely many <i>n<sub>i</sub></i>-by-<i>n<sub>i</sub></i> <a href="Matrix_ring" title="Matrix ring">matrix rings</a> over <a href="Division_ring" title="Division ring">division rings</a> <i>D<sub>i</sub></i>, for some integers <i>n<sub>i</sub></i>, both of which are uniquely determined up to permutation of the index <i>i</i>. In particular, any <a href="Simple_ring" title="Simple ring">simple</a> left or right <a href="Artinian_ring" title="Artinian ring">Artinian ring</a> is isomorphic to an <i>n</i>-by-<i>n</i> <a href="Matrix_ring" title="Matrix ring">matrix ring</a> over a <a href="Division_ring" title="Division ring">division ring</a> <i>D</i>, where both <i>n</i> and <i>D</i> are uniquely determined.<sup id="cite_ref-Beachy1999_5-0" class="reference"><a href="#cite_note-Beachy1999-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>As a direct corollary, the Artin–Wedderburn theorem implies that every simple ring that is finite-dimensional over a division ring (a simple algebra) is a <a href="Matrix_ring" title="Matrix ring">matrix ring</a>. This is <a href="Joseph_Wedderburn" title="Joseph Wedderburn">Joseph Wedderburn</a>'s original result. <a href="Emil_Artin" title="Emil Artin">Emil Artin</a> later generalized it to the case of Artinian rings.

</p>

<div class="mw-heading mw-heading3"><h3 id="Jacobson_density_theorem">Jacobson density theorem</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Jacobson_density_theorem" title="Jacobson density theorem">Jacobson density theorem</a></div>

<p>The <b>Jacobson density theorem</b> is a theorem concerning <a href="Simple_module" title="Simple module">simple modules</a> over a ring <span class="texhtml mvar" style="font-style:italic;">R</span>.<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>

</p><p>The theorem can be applied to show that any <a href="Primitive_ring" title="Primitive ring">primitive ring</a> can be viewed as a "dense" subring of the ring of <a href="Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformations</a> of a vector space.<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><sup id="cite_ref-Isaacs187_8-0" class="reference"><a href="#cite_note-Isaacs187-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by <a href="Nathan_Jacobson" title="Nathan Jacobson">Nathan Jacobson</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> This can be viewed as a kind of generalization of the <a href="Artin-Wedderburn_theorem" class="mw-redirect" title="Artin-Wedderburn theorem">Artin-Wedderburn theorem</a>'s conclusion about the structure of <a href="Simple_ring" title="Simple ring">simple</a> <a href="Artinian_ring" title="Artinian ring">Artinian rings</a>.

</p><p>More formally, the theorem can be stated as follows:

</p>

<dl><dd><b>The Jacobson Density Theorem.</b> Let <span class="texhtml mvar" style="font-style:italic;">U</span> be a simple right <span class="texhtml mvar" style="font-style:italic;">R</span>-module, <span class="texhtml"><i>D</i> = End(<i>U<sub>R</sub></i>)</span>, and <span class="texhtml"><i>X</i> ⊂ <i>U</i></span> a finite and <span class="texhtml mvar" style="font-style:italic;">D</span>-linearly independent set. If <span class="texhtml mvar" style="font-style:italic;">A</span> is a <span class="texhtml mvar" style="font-style:italic;">D</span>-linear transformation on <span class="texhtml mvar" style="font-style:italic;">U</span> then there exists <span class="texhtml"><i>r</i> ∈ <i>R</i></span> such that <span class="texhtml"><i>A</i>(<i>x</i>) = <i>x</i> · <i>r</i></span> for all <span class="texhtml mvar" style="font-style:italic;">x</span> in <span class="texhtml mvar" style="font-style:italic;">X</span>.<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></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Nakayama's_lemma">Nakayama's lemma</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Nakayama's_lemma" title="Nakayama's lemma">Nakayama's lemma</a></div>

<p>Let J(<i>R</i>) be the <a href="Jacobson_radical" title="Jacobson radical">Jacobson radical</a> of <i>R</i>. If <i>U</i> is a right module over a ring, <i>R</i>, and <i>I</i> is a right ideal in <i>R</i>, then define <i>U</i>·<i>I</i> to be the set of all (finite) sums of elements of the form <i>u</i>·<i>i</i>, where <b>·</b> is simply the action of <i>R</i> on <i>U</i>. Necessarily, <i>U</i>·<i>I</i> is a submodule of <i>U</i>.

</p><p>If <i>V</i> is a <a href="Maximal_submodule" class="mw-redirect" title="Maximal submodule">maximal submodule</a> of <i>U</i>, then <i>U</i>/<i>V</i> is <a href="Simple_module" title="Simple module">simple</a>. So <i>U</i>·J(<i>R</i>) is necessarily a subset of <i>V</i>, by the definition of J(<i>R</i>) and the fact that <i>U</i>/<i>V</i> is simple.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Thus, if <i>U</i> contains at least one (proper) maximal submodule, <i>U</i>·J(<i>R</i>) is a proper submodule of <i>U</i>. However, this need not hold for arbitrary modules <i>U</i> over <i>R</i>, for <i>U</i> need not contain any maximal submodules.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> Naturally, if <i>U</i> is a <a href="Noetherian_ring" title="Noetherian ring">Noetherian</a> module, this holds. If <i>R</i> is Noetherian, and <i>U</i> is <a href="Finitely_generated_module" title="Finitely generated module">finitely generated</a>, then <i>U</i> is a Noetherian module over <i>R</i>, and the conclusion is satisfied.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Somewhat remarkable is that the weaker assumption, namely that <i>U</i> is finitely generated as an <i>R</i>-module (and no finiteness assumption on <i>R</i>), is sufficient to guarantee the conclusion. This is essentially the statement of Nakayama's lemma.<sup id="cite_ref-Isaacs_14-0" class="reference"><a href="#cite_note-Isaacs-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup>

</p><p>Precisely, one has the following.

</p>

<dl><dd><b>Nakayama's lemma</b>: Let <i>U</i> be a <a href="Finitely_generated_module" title="Finitely generated module">finitely generated</a> right module over a ring <i>R</i>. If <i>U</i> is a non-zero module, then <i>U</i>·J(<i>R</i>) is a proper submodule of <i>U</i>.<sup id="cite_ref-Isaacs_14-1" class="reference"><a href="#cite_note-Isaacs-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></dd></dl>

<p>A version of the lemma holds for right modules over non-commutative <a href="Unitary_ring" class="mw-redirect" title="Unitary ring">unitary rings</a> <i>R</i>. The resulting theorem is sometimes known as the <b>Jacobson–Azumaya theorem</b>.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Noncommutative_localization">Noncommutative localization</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">Localization of a ring</a></div>

<p>Localization is a systematic method of adding multiplicative inverses to a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>, and is usually applied to commutative rings. Given a ring <i>R</i> and a subset <i>S</i>, one wants to construct some ring <i>R</i>* and <a href="Ring_homomorphism" title="Ring homomorphism">ring homomorphism</a> from <i>R</i> to <i>R</i>*, such that the image of <i>S</i> consists of <i><a href="Unit_(ring_theory)" title="Unit (ring theory)">units</a></i> (invertible elements) in <i>R</i>*. Further one wants <i>R</i>* to be the 'best possible' or 'most general' way to do this – in the usual fashion this should be expressed by a <a href="Universal_property" title="Universal property">universal property</a>. The localization of <i>R</i> by <i>S</i> is usually denoted by <i>S</i><sup> −1</sup><i>R</i>; however other notations are used in some important special cases. If <i>S</i> is the set of the non zero elements of an <a href="Integral_domain" title="Integral domain">integral domain</a>, then the localization is the <a href="Field_of_fractions" title="Field of fractions">field of fractions</a> and thus usually denoted Frac(<i>R</i>).

</p><p>Localizing <a href="Non-commutative_ring" class="mw-redirect" title="Non-commutative ring">non-commutative rings</a> is more difficult; the localization does not exist for every set <i>S</i> of prospective units. One condition which ensures that the localization exists is the <a href="Ore_condition" title="Ore condition">Ore condition</a>.

</p><p>One case for non-commutative rings where localization has a clear interest is for rings of differential operators. It has the interpretation, for example, of adjoining a formal inverse <i>D</i><sup>−1</sup> for a differentiation operator <i>D</i>. This is done in many contexts in methods for <a href="Differential_equation" title="Differential equation">differential equations</a>. There is now a large mathematical theory about it, named <a href="Microlocal_analysis" title="Microlocal analysis">microlocalization</a>, connecting with numerous other branches. The <i>micro-</i> tag is to do with connections with <a href="Fourier_theory" class="mw-redirect" title="Fourier theory">Fourier theory</a>, in particular.

</p>

<div class="mw-heading mw-heading3"><h3 id="Morita_equivalence">Morita equivalence</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Morita_equivalence" title="Morita equivalence">Morita equivalence</a></div>

<p>Morita equivalence is a relationship defined between <a href="Ring_(mathematics)" title="Ring (mathematics)">rings</a> that preserves many ring-theoretic properties. It is named after Japanese mathematician <a href="Kiiti_Morita" title="Kiiti Morita">Kiiti Morita</a> who defined equivalence and a similar notion of duality in 1958.

</p><p>Two rings <i>R</i> and <i>S</i> (associative, with 1) are said to be (<b>Morita</b>) <b>equivalent</b> if there is an equivalence of the category of (left) modules over <i>R</i>, <i>R-Mod</i>, and the category of (left) modules over <i>S</i>, <i>S-Mod</i>. It can be shown that the left module categories <i>R-Mod</i> and <i>S-Mod</i> are equivalent if and only if the right module categories <i>Mod-R</i> and <i>Mod-S</i> are equivalent. Further it can be shown that any functor from <i>R-Mod</i> to <i>S-Mod</i> that yields an equivalence is automatically <a href="Additive_functor" class="mw-redirect" title="Additive functor">additive</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Brauer_group">Brauer group</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Brauer_group" title="Brauer group">Brauer group</a></div>

<p>The Brauer group of a <a href="Field_(mathematics)" title="Field (mathematics)">field</a> <i>K</i> is an <a href="Abelian_group" title="Abelian group">abelian group</a> whose elements are <a href="Morita_equivalence" title="Morita equivalence">Morita equivalence</a> classes of <a href="Central_simple_algebra" title="Central simple algebra">central simple algebras</a> of finite rank over <i>K</i> and addition is induced by the <a href="Tensor_product" title="Tensor product">tensor product</a> of algebras. It arose out of attempts to classify <a href="Division_algebra" title="Division algebra">division algebras</a> over a field and is named after the algebraist <a href="Richard_Brauer" title="Richard Brauer">Richard Brauer</a>. The group may also be defined in terms of <a href="Galois_cohomology" title="Galois cohomology">Galois cohomology</a>. More generally, the Brauer group of a <a href="Scheme_(mathematics)" title="Scheme (mathematics)">scheme</a> is defined in terms of <a href="Azumaya_algebra" title="Azumaya algebra">Azumaya algebras</a>.

</p>

<div class="mw-heading mw-heading3"><h3 id="Ore_conditions">Ore conditions</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Ore_condition" title="Ore condition">Ore condition</a></div>

<p>The Ore condition is a condition introduced by <a href="%C3%98ystein_Ore" title="Øystein Ore">Øystein Ore</a>, in connection with the question of extending beyond <a href="Commutative_ring" title="Commutative ring">commutative rings</a> the construction of a <a href="Field_of_fractions" title="Field of fractions">field of fractions</a>, or more generally <a href="Localization_of_a_ring" class="mw-redirect" title="Localization of a ring">localization of a ring</a>. The <i>right Ore condition</i> for a <a href="Multiplicative_subset" class="mw-redirect" title="Multiplicative subset">multiplicative subset</a> <i>S</i> of a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> <i>R</i> is that for <span class="nowrap"><i>a</i> ∈ <i>R</i></span> and <span class="nowrap"><i>s</i> ∈ <i>S</i></span>, the intersection <span class="nowrap"><i>aS</i> ∩ <i>sR</i> ≠ ∅</span>.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> A domain that satisfies the right Ore condition is called a <b>right Ore domain</b>. The left case is defined similarly.

</p>

<div class="mw-heading mw-heading3"><h3 id="Goldie's_theorem">Goldie's theorem</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Goldie's_theorem" title="Goldie's theorem">Goldie's theorem</a></div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, <b>Goldie's theorem</b> is a basic structural result in <a href="Ring_theory" title="Ring theory">ring theory</a>, proved by <a href="Alfred_Goldie" title="Alfred Goldie">Alfred Goldie</a> during the 1950s. What is now termed a right <b>Goldie ring</b> is a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> <i>R</i> that has finite <a href="Uniform_dimension" class="mw-redirect" title="Uniform dimension">uniform dimension</a> (also called "finite rank") as a right module over itself, and satisfies the <a href="Ascending_chain_condition" title="Ascending chain condition">ascending chain condition</a> on right <a href="Annihilator_(ring_theory)" title="Annihilator (ring theory)">annihilators</a> of subsets of <i>R</i>.

</p><p>Goldie's theorem states that the <a href="Semiprime_ring" title="Semiprime ring">semiprime</a> right Goldie rings are precisely those that have a <a href="Semisimple_ring" class="mw-redirect" title="Semisimple ring">semisimple</a> <a href="Artinian_ring" title="Artinian ring">Artinian</a> right <a href="Classical_ring_of_quotients" class="mw-redirect" title="Classical ring of quotients">classical ring of quotients</a>. The structure of this ring of quotients is then completely determined by the <a href="Artin%E2%80%93Wedderburn_theorem" class="mw-redirect" title="Artin–Wedderburn theorem">Artin–Wedderburn theorem</a>.

</p><p>In particular, Goldie's theorem applies to semiprime right <a href="Noetherian_ring" title="Noetherian ring">Noetherian rings</a>, since by definition right Noetherian rings have the ascending chain condition on <i>all</i> right ideals. This is sufficient to guarantee that a right-Noetherian ring is right Goldie. The converse does not hold: every right <a href="Ore_domain" class="mw-redirect" title="Ore domain">Ore domain</a> is a right Goldie domain, and hence so is every commutative <a href="Integral_domain" title="Integral domain">integral domain</a>.

</p><p>A consequence of Goldie's theorem, again due to Goldie, is that every semiprime <a href="Principal_right_ideal_ring" class="mw-redirect" title="Principal right ideal ring">principal right ideal ring</a> is isomorphic to a finite direct sum of <a href="Prime_ring" title="Prime ring">prime</a> principal right ideal rings. Every prime principal right ideal ring is isomorphic to a <a href="Matrix_ring" title="Matrix ring">matrix ring</a> over a right Ore domain.

</p>

<ul><li><a href="Derived_algebraic_geometry" title="Derived algebraic geometry">Derived algebraic geometry</a></li>

<li><a href="Noncommutative_geometry" title="Noncommutative geometry">Noncommutative geometry</a></li>

<li><a href="Noncommutative_algebraic_geometry" title="Noncommutative algebraic geometry">Noncommutative algebraic geometry</a></li>

<li><a href="Noncommutative_harmonic_analysis" title="Noncommutative harmonic analysis">Noncommutative harmonic analysis</a></li>

<li><a href="Representation_theory_(group_theory)" class="mw-redirect" title="Representation theory (group theory)">Representation theory (group theory)</a></li></ul>

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

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</style><cite id="CITEREFSloane_"A127708"" class="citation web cs1"><a href="Neil_Sloane" title="Neil Sloane">Sloane, N. J. A.</a> (ed.). <a rel="nofollow" class="external text" href="https://oeis.org/A127708">"Sequence A127708 (Number of non-commutative rings with 1)"</a>. <i>The <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a></i>. OEIS Foundation.</cite></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">In this article, rings have a 1.</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"><cite id="CITEREFShult2011" class="citation book cs1">Shult, Ernest E. (2011). <i>Points and lines. Characterizing the classical geometries</i>. Universitext. Berlin: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>. p. 123. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-642-15626-7</bdi>. <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1213.51001">1213.51001</a>.</cite></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"><a href="Semisimple_ring" class="mw-redirect" title="Semisimple ring">Semisimple rings</a> are necessarily <a href="Artinian_ring" title="Artinian ring">Artinian rings</a>. Some authors use "semisimple" to mean the ring has a trivial <a href="Jacobson_radical" title="Jacobson radical">Jacobson radical</a>. For Artinian rings, the two notions are equivalent, so "Artinian" is included here to eliminate that ambiguity.</span>

</li>

<li id="cite_note-Beachy1999-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Beachy1999_5-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFJohn_A._Beachy1999" class="citation book cs1">John A. Beachy (1999). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/introductorylect0000beac"><i>Introductory Lectures on Rings and Modules</i></a></span>. Cambridge University Press. p. <a rel="nofollow" class="external text" href="https://archive.org/details/introductorylect0000beac/page/156">156</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-64407-5</bdi>.</cite></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">Isaacs, p. 184</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">Such rings of linear transformations are also known as <a href="Full_linear_ring" class="mw-redirect" title="Full linear ring">full linear rings</a>.</span>

</li>

<li id="cite_note-Isaacs187-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Isaacs187_8-0">^</a></b></span> <span class="reference-text">Isaacs, Corollary 13.16, p. 187</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"><a href="#CITEREFJacobson1945">Jacobson 1945</a></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">Isaacs, Theorem 13.14, p. 185</span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><a href="#CITEREFIsaacs1993">Isaacs 1993</a>, p. 182</span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><a href="#CITEREFIsaacs1993">Isaacs 1993</a>, p. 183</span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><a href="#CITEREFIsaacs1993">Isaacs 1993</a>, Theorem 12.19, p. 172</span>

</li>

<li id="cite_note-Isaacs-14"><span class="mw-cite-backlink">^ <a href="#cite_ref-Isaacs_14-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Isaacs_14-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="#CITEREFIsaacs1993">Isaacs 1993</a>, Theorem 13.11, p. 183</span>

</li>

<li id="cite_note-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-15">^</a></b></span> <span class="reference-text"><a href="#CITEREFNagata1962">Nagata 1962</a>, §A2</span>

</li>

<li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><cite id="CITEREFCohn1991" class="citation book cs1">Cohn, P. M. (1991). "Chap. 9.1". <i>Algebra</i>. Vol. 3 (2nd ed.). p. 351.</cite></span>

</li>

</ol></div>

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

<ul><li><cite id="CITEREFIsaacs1993" class="citation cs2"><a href="Martin_Isaacs" class="mw-redirect" title="Martin Isaacs">Isaacs, I. Martin</a> (1993), <i>Algebra, a graduate course</i> (1st ed.), <a href="Brooks/Cole" class="mw-redirect" title="Brooks/Cole">Brooks/Cole</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-534-19002-2</bdi></cite></li>

<li><cite id="CITEREFJacobson1945" class="citation cs2"><a href="Nathan_Jacobson" title="Nathan Jacobson">Jacobson, N.</a> (1945), "Structure theory of simple rings without finiteness assumptions", <i><a href="Transactions_of_the_American_Mathematical_Society" title="Transactions of the American Mathematical Society">Transactions of the American Mathematical Society</a></i>, <b>57</b>: <span class="nowrap">228–</span>245, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1990204">10.2307/1990204</a></span>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/1990204">1990204</a></cite></li>

<li><cite id="CITEREFNagata1962" class="citation cs2"><a href="Masayoshi_Nagata" title="Masayoshi Nagata">Nagata, M.</a> (1962), <i>Local Rings</i>, <a href="Wiley-Interscience" class="mw-redirect" title="Wiley-Interscience">Wiley-Interscience</a></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite class="citation cs2"><a href="Israel_Nathan_Herstein" title="Israel Nathan Herstein">Herstein, I. N.</a> (1968), <i>Noncommutative Rings</i>, <a href="Mathematical_Association_of_America" title="Mathematical Association of America">Mathematical Association of America</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-88385-015-X</bdi></cite></li>

<li><cite class="citation cs2"><a href="Tsit_Yuen_Lam" title="Tsit Yuen Lam">Lam, T. Y.</a> (2001), <i>A First Course in Noncommutative Rings</i>, <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a></cite></li></ul>

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