Suppose you have the following conditons: $f(x+y) = f(x) + f(y), f(xy) = f(x)f(y)$ and $f(1) = 0$, for all $x, y$. Would these conditions be sufficient to form a ring homomorphism always or is there a counterexample? I've been trying to show that you automatically get $f(1) = 1$ from $f(xy) = f(x)f(y)$, but I can't seem to find a counterexample that eliminates the $f(1) = 0$ case. Any help would be greatly appreciated.
When you say that f(1)(1-f(1))=0 implies f(1)=0 or f(1)=1 you are implicitly assuming that ab=0 implies a=0 or b=0. This is not true in every ring. The rings for which ab=0 implies a=0 or b=0 are called "integral domains" or just "domains" for short. For non-trivial ring homomorphisms, the unity-preserving axiom is necessary to include.