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. It has the advantage of adding a lot more structure to

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 subring (not necessarily with identity) closed under multiplication by arbitrary ring elements is called an ideal. These relate to the kernels of ring HomoMorphisms the same way NormalSubGroups? relate to those of GroupHomomorphism?s.

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, IntegralDomain?s where every non-zero element has a muliplicative inverse.

Every ring contains a unique smallest subring (with identity), isomorphic to either the IntegerNumbers or a ModularArithmetic. Every field containing the IntegerNumbers contains a unique smallest field isomorphic to the RationalNumbers.