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# ring

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1 An algebraic structure R on which there are defined two dyadic operations, normally denoted by + (addition) and · or juxtaposition (multiplication). With respect to addition, R is an abelian group,〈R, +〉i.e. + is commutative and associative. With respect to multiplication, R is a semigroup,〈R, ·〉i.e. · is associative. Further, multiplication is distributive over addition.

R, +〉

R, ·〉

Certain kinds of rings are of particular interest:(a) if multiplication is commutative the ring is called a commutative ring;(b) if 〈R, ·〉 is a monoid, the ring is called a ring with an identity;(c) a commutative ring with an identity, and having no nonzero elements x and y with the property that x · y = 0, is said to be an integral domain;(d) a commutative ring with more than one element, and in which every nonzero element has an inverse with respect to multiplication, is called a field.The different identity elements and inverses, when these exist, can be distinguished by talking in terms of additive identities (or zeros), multiplicative identities (or ones), additive inverses, and multiplicative inverses.

(a) if multiplication is commutative the ring is called a commutative ring;

(b) if 〈R, ·〉 is a monoid, the ring is called a ring with an identity;

(c) a commutative ring with an identity, and having no nonzero elements x and y with the property that x · y = 0, is said to be an integral domain;

(d) a commutative ring with more than one element, and in which every nonzero element has an inverse with respect to multiplication, is called a field.

The concept of a ring provides an algebraic structure into which can be fitted such diverse items as the integers, polynomials with integer coefficients, and matrices; on all these items it is customary to define two dyadic operations.

2Another name for circular list, but more generally applied to any list structure where all sublists as well as the list itself are circularly linked.

3 In network topology, a ring network is a closed-loop network that does not require terminators. A token ring topology is physically cabled as a star, with a logical ring maintained at the hub. When a workstation connects to the hub, the ring is extended out to the workstation and back to the hub.

Subjects: Computing.

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