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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.
Algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of commutative algebra, a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and commutative ...
By convention, a ring has the multiplicative identity. But some authors do not require a ring to have the multiplicative identity; i.e., for them, a ring is a rng. For a rng R, a left ideal I is a subrng with the additional property that is in I for every and every . (Right and two-sided ideals are defined similarly.)
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra . Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings.
Let be a group, written multiplicatively, and let be a ring. The group ring of over , which we will denote by [], or simply , is the set of mappings : of finite support (() is nonzero for only finitely many elements ), where the module scalar product of a scalar in and a mapping is defined as the mapping (), and the module group sum of two mappings and is defined as the mapping () + ().
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring.The concept of a module also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers.
In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a simple ring is necessarily a field. It follows that a simple ring is an associative algebra over this
In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.