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.

 \begin{gathered} \left( i \right)\,a + b \in  R\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[  {{\text{Closure}}\,{\text{Law}}\,{\text{for}}\,{\text{Addition}}} \right] \\ \left( {ii} \right)\left( {a + b} \right) + c  = a + \left( {b + c}  \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[  {{\text{Associative}}\,{\text{Law}}\,{\text{of}}\,{\text{Addition}}} \right] \\ \left( {iii} \right)a + 0 = a = 0 + a\,\,\,\forall  a \in R\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {{\text{Existence  of Additive Identity}}} \right] \\ \left( {iv} \right)\,a + \left( { - a}  \right) = - a + a = 0\,\,\,\forall a \in  R\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {{\text{Existence of Additive Inverse}}}  \right] \\ \left( v \right)\,a + b = b + a\,\,\,\forall  a \in  R\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[  {{\text{Commutative}}\,{\text{Law}}\,{\text{of}}\,{\text{Addition}}} \right] \\<br />
\end{gathered}


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

\begin{gathered} \left( i \right)\,a \cdot b \in  R\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[  {{\text{Closure}}\,{\text{Law}}\,{\text{for}}\,{\text{Multiplication}}} \right] \\ \left( {ii} \right)\left( {a \cdot b} \right)  \cdot c = a \cdot \left( {b \cdot c}  \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left[ {{\text{Associative}}\,{\text{Law}}\,{\text{of}}\,{\text{Multiplication}}}  \right] \\ \end{gathered}


{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