Books give different definitions for an
integral domain. Namely:
A commutative ring D that has an additional axiom, the cancellation law, which is:
for any a, b, c in a commutative ring D, ab = ac and a != 0 implies b = c.
(1)
And as
Noether points out:
A commutative ring D that has the property: for any a, b in D, ab = 0 implies a = 0 or b = 0.
(2)
The definitions are equivalent, and it is easy to show. (Upon request, I'll post the proof that says they are equivalent definitions.)
The first definition emphasizes the multiplicative cancellation law, while the second emphasizes a domain's usefulness in finding roots.