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Noncommutative ring
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a noncommutative ring is a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> whose multiplication is not <a href="/facts/Commutative_property/WaYxJLxd">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="/facts/Commutative_ring/QS1r2ovR">commutative ring</a>. </p><p>Noncommutative algebra is the part of <a href="/facts/Ring_theory/VW6svEFe">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>