Determine all ring homomorphisms of $\displaystyle \mathbb{Z} x \mathbb{Z} $ into $\displaystyle \mathbb{Z} x \mathbb{Z} $.
Clearly justify your method.
Since every element in ZxZ can be written as m(1,0) + n(0,1) for m,n in $\displaystyle Z$ 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.