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.


(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.