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In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers.
Sometimes the term noncommutative ring is used instead of ring 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, ring is used as a shorthand for commutative ring.
Given a module M over a ring R, an R endomorphism f of M is called an involution if f 2 is the identity homomorphism on M. Involutions are related to idempotents; if 2 is invertible then they correspond in a one-to-one manner. In functional analysis, Banach *-algebras and C*-algebras are special types of Banach algebras with involutions.
The nilpotent elements of a commutative ring R form an ideal of R, called the nilradical of R; therefore a commutative ring is reduced if and only if its nilradical is zero. Moreover, a commutative ring is reduced if and only if the only element contained in all prime ideals is zero. A quotient ring R/I is reduced if and only if I is a radical ...
Ring sideroblasts are named so because iron-laden mitochondria form a ring around the nucleus. ... Acquired reversible sideroblastic anemia.
The Warriors rings are huge, gaudy and ... reversible.
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