mod should be part of the signature of the ring, so you are not looking at pure multiplication. Then the domain would be still integral. Why wouldn't it?
An integral domain is any commutative ring with no zero divisors (a,b are zero divisors if ab = 0 but a != 0, b != 0). In this case 2*2 = 0, so 2 is a zero divisor, so Z/4z = Z_4 is not an integral domain.