I got $(a,b) \to a$, $(a,b) \to b$ and $(a,b) \to 0$ these mappings to be homomorphisms just by hit and trial. So when I looked for it's solution, these were the ONLY homomorphisms from $ \Bbb Z$ $\oplus$ $\Bbb Z $ to $\Bbb Z$.
There was a hint, which is as follows: "Observe that an idempotent must map to an idempotent."
Does this always happen? Why in this case it happens? (EDIT)How should I approach to construct those three homomorphisms by using the hint?
This problem belongs to the book Contemporary abstract algebra by Gallian.