Commutative Ring

Definition

A ring in which the multiplication is commutative is a commutative ring. A ring with a multiplicative identity element is a ring with unity; the multiplicative identity element $1$ is called “unity”.

In a ring with unity $1$ the distributive laws show that

$(\underset{n\text{ summands}}{\underbrace{1+1+\cdots +1}})(\underset{m\text{ summands}}{\underbrace{1+1+\cdots +1}})=(\underset{nm\text{ summands}}{\underbrace{1+1+\cdots +1}})$

that is, $(n\cdot 1)(m\cdot 1)=(nm)\cdot 1$