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.
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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

Except for the terms you use to define 'ring'; those are more primitive than 'ring', I hope! -- LarrySanger