I need help proving the following:

1. Let be a Noetherian ring, an ideal, and be -modules. Let be a finitely generated -module. Prove that is a finitely generated -module.

2. Let be a Noetherian ring, an ideal of R. Prove that is a finitely generated -module iff is a finitely generated -module.

The first problem seems to be straightforward. Can I just take the generators and multiply them by to show this is finitely generated? Thanks for any suggestions with this one.

As for the second one, I am not seeing how to use the first part in this problem. I am guessing that I just let for one direction; but the other direction is confusing me.