RinG

A Ring is a commutative group under an operation +, together with a second operation * s.t.

a*(b*c) = (a*b)*c

a*(b+c) = (a*b)+(a*c), (a+b)*c = (a*c)+(b*c)

Very often the definition of a ring is taken to require a multiplicative identity, or unity, denoted 1. Nearly all important rings actually satisfy this. It has the disadvantage, however, of making ring ideals not subrings, as compared with their group-analog, the normal subgroups.

Some important concepts: SubRings (including ideals)

Sets of commutative group HomoMorphisms form rings under addition and composition, provided they are closed under all necessary operations. An isomorphism from a ring to such a collection is called a representation of the ring, and groups under ring-actions are referred to as ModulEs. Every ring has some sort of faithful representation.

A ring where no two non-zero elements multiply to give zero is called an IntegralDomain. In such rings, multiplicative cancellation is possible. Of particular interest are FielDs, IntegralDomains where every non-zero element has a muliplicative inverse.