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.