A is the set Z (integers), with the following "addition"+and "multiplication" * :

a + b = a + b - 1 and a * b = ab - (a + b) + 2

I've already proven that A is a commutative ring with unity. Now, I have to prove that A is an integral domain.

To my understanding, in order to do this, I assume that a * b = 0 and show that either a = 0 or b = 0. Please correct me if I'm wrong...

So, a * b = ab - (a + b) + 2 = ab - a - b + 2 = 0.

I can't figure out how to proceed and show that either a or b equals zero. Can someone help here? Thanks!