(Solved): 11. Let f:RS be a ring homomorphism, J an ideal of S and I={rRf(r)J}. Prove that I is a ...

11. Let $f:R⟶S$ be a ring homomorphism, $J$ an ideal of $S$ and $I=\{r\in R\mid f(r)\in J\}$. Prove that $I$ is an ideal of $R$ that contains the kernel of $f$ [Hint: first show that $I$ is a subring of $R$ ] 12. In a finite commutative ring $R$ with identity, prove that an ideal is maximal if and only if it is prime