Ring of Integers Isomorphism Problem

e(ho0n3

The problem statement, all variables and given/known data
Let N = AB, where A and B are positive integers that are relatively prime. Prove that Z_N is isomorphic to Z_A x Z_B.

The attempt at a solution
I'm considering the map f(n) = (n mod A, n mod B). I've been able to prove that it is homomorphic and injective. Is it safe to conclude, since Z_N and Z_A x Z_B have the same cardinality and f is injective, that f is surjetive? In any case, given an (a, b) in Z_A x Z_B, I've been trying to find an n such that f(n) = (a, b) without success. Any tips?

morphism

Recall the Chinese remainder theorem.

e(ho0n3

Good tip. Thanks.