RinGs

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

1. a*(b*c) = (a*b)*c

1. 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

A justification for the definition of a ring is that collections of commutative group homomorphisms closed under composition form rings (much the same way bijections form groups). The commutative group is referred to as a module under the ring; the module is called faithful if the actions of ring elements are each different.

A ring where no two non-zero elements multiply to give zero is called an IntegralDomain. The most interesting of these are FielDs, where every non-zero element has a multiplicative inverse.

Some important rings include the integers, modular arithmetics, and polynomials.

Why not define some of the simpler terminology here and used in other mathematical articles? Would be interesting to wikify an entire mathematico-deductive system of definitions in this way. -- LarrySanger

I would say groups, rings, etc are the simpler terminology, since everything else are special cases. :) -JG