Algebraic structure also called skew field

<span class="mw-page-title-main">Division ring</span>

Division ring

<p>In <a href="/en/Algebra" title="Algebra" class="wl">algebra</a>, a <b>division ring</b>, also called a <b>skew field</b>, is a <a href="/en/Zero_ring" title="Zero ring" class="wl">nontrivial</a> <a href="/en/Ring_(mathematics)" title="Ring (mathematics)" class="wl">ring</a> in which <a href="/en/Division_(mathematics)" title="Division (mathematics)" class="wl">division</a> by nonzero elements is defined. Specifically, it is a nontrivial ring<span class="mw-ref reference" id="cite_ref-1"><a href="#cite_note-1" style="counter-reset: mw-Ref 1;"><span class="mw-reflink-text">[1]</span></a></span> in which every nonzero element <span class="texhtml mvar" style="font-style:italic;">a</span> has a <a href="/en/Multiplicative_inverse" title="Multiplicative inverse" class="wl">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> </span></span>a</i><sup>–1</sup> = <i>a</i><sup>–1</sup><span class="nowrap"><span> </span></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> </span></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> </span></span>b</i><sup>–1</sup> ≠ <i>b</i><sup>–1</sup><span class="nowrap"><span> </span></span><i>a</i></span>.</p>

<div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none" about="#mwt1">Algebraic structure also called skew field</div>
<p>A commutative division ring is a <a href="/en/Field_(mathematics)" title="Field (mathematics)" class="wl">field</a>. <a href="/en/Wedderburn's_little_theorem" title="Wedderburn's little theorem" class="wl">Wedderburn's little theorem</a> asserts that all finite division rings are commutative and therefore <a href="/en/Finite_field" title="Finite field" class="wl">finite fields</a>. </p>

<p>Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields".<span class="mw-ref reference" id="cite_ref-5"><a href="#cite_note-5" style="counter-reset: mw-Ref 5;"><span class="mw-reflink-text">[5]</span></a></span> In some languages, such as <a href="/en/French_language" title="French language" class="wl">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="/en/Simple_ring" title="Simple ring" class="wl">simple</a>. That is, they have no two-sided <a href="/en/Ideal_(ring_theory)" title="Ideal (ring theory)" class="wl">ideal</a> besides the <a href="/en/Zero_ideal" title="Zero ideal" class="mw-redirect wl">zero ideal</a> and itself.</p>

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In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element a has a multiplicative inverse, that is, an element usually denoted a–1, such that a a–1 = a–1 a = 1. So, division may be defined as a / b = a b–1, but this notation is avoided, as one may have a b–1 ≠ b–1 a. A commutative division ring is a field. Wedderburn's little theorem asserts that all finite division rings are commutative and therefore finite fields.  Historically, division rings were sometimes referred to as fields, while fields were called "commutative fields". In some languages, such as French, the word equivalent to "field" is used for both commutative and noncommutative cases, and the distinction between the two cases is made by adding qualificatives such as "corps commutatif" or "corps gauche" . 