# Introduction to Rings in Algebra

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.

Definition:
An algebraic structure $\left( {R, + , \times } \right)$ where $R$ is a non-empty set and $+$ and $\times$ are defined operations in $R$, is called a ring if for all $a,b,c$ in $R$, the following axioms are satisfied:
${R_1}:$ $\left( {R, + } \right)$ is an abelian group.

${R_2}:$  $\left( {R, \times } \right)$ is an semi group.

${R_3}:$ Multiplication is left as well as right distributive over addition, i.e.
$a \cdot \left( {b + c} \right) = a \cdot b + a \cdot c$ and $\left( {b + c} \right) \cdot a = b \cdot a + c \cdot a$