1. The problem statement, all variables and given/known data Suppose that f:R->Q (reals to rationals) is a ring homomorphism. Prove that f(x)=0 for every x in the reals. 2. Relevant equations Homomorphisms map the zero element to the zero element. f(0) = 0 Homomorphisms preserve additive inverses. f(-a)=-f(a) and finally, f(a - b) = f(a) - f(b) 3. The attempt at a solution My guess is go with contradiction and say, Suppose that f(a != 0) != 0 for some a in the reals. But I don't see where to go from there. A hint or suggestion would be nice.