integral domain for set of integers

Question

In an integral domain, a product is zero only when one of the factors is zero; that is, ab=0 only when a=0 or b=0.
Can we do the following steps for proving set of integers an integral domain?
let a=3 and b=0
for all a,b belongs to integers
a.b=3.0=0
If not kindly show how it can be done.

Answer 1

You have shown that there are integers $a$ and $b$ such that $a.b=0$.

You have not shown that if $a.b=0$ then $a=0$ or $b=0$.