Let R be a ring and f: R → R be defined by f(x) = x³. Check All that are correct. fis onto when R = Zg. fis a group homomorphism for (R = Z5,+). f is one-to-one when R = Z7. f is not onto when R= Z3. fis a ring homomorphism for (R = Z3,+, . ). f is a group homomorphism for (R = Z3,+).

Determine if each of the following maps is a ring homomorphism. (a) f : R → R defined by f(x) %3D -x 1 -2 = (6 1) a ( (b) g : M2(IR) → M2(R) defined 1

5. G (a + b.2: a eZ and b e Z) is a group under addition in the reals. Define o : G -→G for all a + b.2 e G, as 0(a+ b.2) = a- b./2. Prove that o is an automorphism.

5. Let R be a commutative ring with identity. Let f: R[x] - f(p(x)) = E, ai, where p(x) = ao+a1x + a2x² + ... + amx™m. Prove that f is a homomorphism of rings. Notice that f(p(x)) = p(1), where 1 = 1R E R. → R be defined by 4 4. Risc