Cryptography: Zn is more than a commutative ring, but not quite an integral domain. Why?

Because Zn contains a multiplicative identity element. Commutative rings are not required to possess multiplicative identities.

Why is Zn not an integral domain?
Even though Zn possesses a multiplicative identity, it does NOT satisfy the other condition of integral domains which says that if a × b = 0 then either a or b must be zero.
Consider modulo 8 arithmetic. We have 2 × 4 = 0, which is a clear violation of the second rule for integral domains.