Let $f: (A,+,.) \rightarrow (B,+,.)$ is surjective ring homomorphism.
Prove or disprove:
(a) If B is commutative ring then A is also commutative.
(b) If B has a multiplicative identity then A has also multiplicative identity.
I know that the inverse implications are true and proof is not difficult. I think that these implications (a) and (b) are false (because it is true when $f$ is bijective, there is equivalence in these claims ). But I cannot find any contra example, or I cannot show that it is not true.
Thanks for any help.