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.
A ring is commutative if its multiplication is commutative.
Some important concepts: subrings and ideals, Modules
A ring where no two non-zero elements multiply to give zero is called an [integral domain]?. In such rings, multiplicative cancellation is possible. Of particular interest are fields, integral domains where every non-zero element has a muliplicative inverse.
Somebody please rename this to mathematical ring!
/Talk?