Integral Domain: A commutative ring with unity is said to be an integral domain if it has no zero-divisors. Alternatively a commutative ring with unity is called an integral domain if for all , .
(i) The set of integers under usual addition and multiplication is an integral domain for any two integers , .
(ii) Consider a ring under the addition and multiplication modulo 8. This ring is commutative but it is not an integral domain because , are two non-zero elements such that .
(iii) The ring of complex numbers is an integral domain.
Let . It is easy to prove that is a commutative ring with unity. The zero element is and unit element is . Also this ring is free from zero-divisor because the product of two non-zero complex numbers cannot be zero. Hence is an integral domain.