Integral Domain in Rings

Integral Domain: A commutative ring with unity is said to be an integral domain if it has no zero-divisors. Alternatively a commutative ring $R$ with unity is called an integral domain if for all $a,b \in R$, $ab = 0 \Rightarrow a = 0\,\,\,{\text{or}}\,\,\,b = 0$.

Examples:

(i) The set $I$ of integers under usual addition and multiplication is an integral domain for any two integers $a,b$, $ab = 0 \Rightarrow a = 0\,\,\,{\text{or}}\,\,\,b = 0$.

(ii) Consider a ring $R = \left\{ {0,1,2,3,4,5,6,7} \right\}$ under the addition and multiplication modulo 8. This ring is commutative but it is not an integral domain because $2 \in R$, $4 \in R$ are two non-zero elements such that $2 \cdot 4 \equiv 0\left( {\bmod 8} \right)$.

(iii) The ring of complex numbers $\mathbb{C}$ is an integral domain.

Let $J\left( i \right) = \left\{ {a + ib:a,b \in I} \right\}$. It is easy to prove that $J\left( i \right)$ is a commutative ring with unity. The zero element is $0 + 0i$ and unit element is $1 + 0i$. Also this ring is free from zero-divisor because the product of two non-zero complex numbers cannot be zero. Hence $J\left( i \right)$ is an integral domain.