Math Help - Ring Homomorphisms

1. Ring Homomorphisms

Determine all ring homomorphisms of $\mathbb{Z} x \mathbb{Z}$ into $\mathbb{Z} x \mathbb{Z}$.

Since every element in ZxZ can be written as m(1,0) + n(0,1) for m,n in $Z$ I think every homomorphism would send [ (1,0) to (n,0), (0,1) to (0,m) ]for integers n, m and then theres just the identity homomorphism.