Introduction to Rings in Algebra

The concept of a group has its origin in the set of mappings or permutations of a set unto 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 addition and multiplication. Thus we now aim at studying rings which are algebraic systems with two suitably restricted and related binary operations.

 

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 a 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