Rings

In mathematics, a ring is an algebraic structure which generalizes the algebraic properties of the integers. Rings, unlike groups, contain two operations usually called addition and multiplication. The branch of abstract algebra which studies rings is called ring theory. more...

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Motivation

In mathematics, objects commonly arise which have structure similar to the integers, but may behave differently in some ways. For example, matrices can be added and multiplied as expected, but such multiplication does not in general satisfy the commutative law. As a different example, the integers modulo n satisfy similar laws of arithmetic but have zero divisors if n is not prime.

A ring is an abstraction of certain properties of the integers that is general enough to allow the study of a greater variety of objects, but strong enough to ensure a rich theory in which substantial results can be proven. In a sense, rings have more structure than an abelian group but less than a field.

Formal definition

A ring is a set + is associative); + is commutative); As with groups the symbol · is usually omitted and multiplication is just denoted by juxtaposition. Also, the standard order of operation rules are used, so that, for example, a+bc is an abbreviation for a+(b·c).

Read more at Wikipedia.org