There are still others, for example, .
Instead of just trying to come up with examples of homomorphisms, begin with the assumption that is a homomorphism and see what you can deduce from that assumption alone.
Find all the homomorphisms from Z -> Z x Z and Z x Z -> Z.
For the first, I've found and proven the following.
Z -> Z x Z such that a -> (a,0) or (0,a) or (0,0).
For the second one:
Z x Z -> Z such that (a,b)-> a or b (the "forgetful" function) or 0.
So I have found three. Am I on the right track? I have verified that these are indeed a homomorphism.