1. Let f: R --> S be a ring homomorphism

Let J be an ideal of S. Prove that is an ideal

of R.

I can see how to show this is true for an isomorphism but not for an homomorphism.

I think there may be an error in the question. If the homomorphism is not bijective (thus is not an isomorphism) than it is possible that doesn't exist. In this case, the statement would not be true. Am I right?

2. Prove that if f is surjective and I is an ideal of R then f(I) is an ideal.

Let

We know . Therefore, because f is an homomorphism, , and

We have and because f is an homomorphism thus

We have and because f is an homomorphism and that gives and conversely s

Ok I miss some steps because it is tidious to write but you understand what I did. I don't understand where the fact that f is surjective intervene. Can you tell me where and why please?