The concept of a group has its origin in the set of mappings or permutations, of a set onto itself. So far we have considered sets with one binary operation only. But rings are the motivation which arises from the fact that integers follow a definite pattern with respect to the addition and multiplication. Thus we now aim at studying rings which are algebraic systems with two suitably restricted and related binary operation.
An algebraic structure where is a non-empty set and and are defined operations in , is called a ring if for all in , the following axioms are satisfied:
is an abelian group.
is an semi group.
Multiplication is left as well as right distributive over addition, i.e.