a) Let R and S be any rings. Show that the zero map, (ie, f: RS defined by f(r) = 0s for all rER) is a ring homomorphism. b) Show Z → M (Z) defined by f(x) = [02] is a ring homomorphism. c) Prove that if f: RS is any ring homomorphism, f(0R) = 0s. d) Let f: Z16 Z4 x Z4 be a ring homomorphism. By (c), f(0)=(0,0). If f(1) = (1,1), find the image of the other elements of Z16. Is f injective? Is f surjective? Justify your answers.

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