I need help with these two questions:
1. Let be a Noetherian ring, an ideal, and be -modules. Suppose is reduced and are the minimal primes of . Prove that is a finitely generated -module iff is a finitely generated -module for each .
2. Now suppose is Artinian (need not be reduced). Prove that is Noetherian iff is Artinian.
I know how to do #1 . But, I don't see how to do #1 . Also, for #2, I am stuck on both implications right now. Thanks in advance.