"Let F be a field and let f:F->R be a ring homomorphism satisfying f(0) != f(1). Show that f is necessarily injective." Assume f(a)=f(b), then f(a)-f(b)=0R => f(a-b)=0R. f(0F)=0R and therefore a=b. But this implies that every homomorphism is injective. How can that be?