I need to prove that there exist no nontrivial homomorphisms from Z to Z. ( Trivial homomorphisms being f(x) = 0 and f(x) = x ) Solution If f is a ring homomorphism from Z to Z then f(1) = f(1*1) = f(1) * f(1) So we have 0 = f(1) * f(1) - f(1) 0 = f(1) * [f(1) - 1] Since Z has no zero divisors, we must have f(1) = 0 or f(1) = 1. In the former case f is the trivial homomorphism f(x) = 0. In the latter, f is the trivial homomorphism f(x) = x, since it is fairly easy to convince yourself that for any x in Z we have f(x) = x * f(1). .