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May 4th, 2009, 11:24 PM   #1
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Noetherian ring, finitely generated module

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.
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