# 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] \\

\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 $$\left( {b + c} \right) \cdot a = b \cdot a + c \cdot a$$

andrew makatu

July 23@ 6:56 pmits enjoying how a ring comes about: from a set to a group (4 properties) from group to a ring (8 properties) and then a field