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 May 4th, 2009, 10:24 PM #1 Newbie   Joined: Nov 2008 Posts: 2 Thanks: 0 Noetherian ring, finitely generated module I need help with these two questions: 1. Let $R$ be a Noetherian ring, $I$ an ideal, and $N \subseteq M$ be $R$-modules. Suppose $R$ is reduced and $P_1, \ldots, P_n$ are the minimal primes of $R$. Prove that $M$ is a finitely generated $R$-module iff $\frac{M}{P_iM}$ is a finitely generated $\frac{R}{P_i}$-module for each $i=1, \ldots, n$. 2. Now suppose $R$ is Artinian (need not be reduced). Prove that $M$ is Noetherian iff $M$ is Artinian. I know how to do #1 $(\Rightarrow)$. But, I don't see how to do #1 $(\Leftarrow)$. Also, for #2, I am stuck on both implications right now. Thanks in advance.

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