i just need a counter example, this has been bothering me all day...!

If R and S are isomorphic commutative rings, then any ring homomorphism $\displaystyle f: R \to S$ is an isomorphism.

I wanted to say....

if $\displaystyle m \geq 2$, then the map $\displaystyle f:\mathbb{Z} \to \mathbb{Z}_m, $ given by $\displaystyle f(n) = [n]$, is not injective, but it is surjective....... but my two rings don't seem to be isomophic..