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 April 29th, 2009, 10:01 PM #1 Newbie   Joined: Dec 2008 Posts: 10 Thanks: 0 Noetherian, finitely generated R-module I need help proving the following: 1. Let $R$ be a Noetherian ring, $I$ an ideal, and $N \subseteq M$ be $R$-modules. Let $\frac{M}{N}$ be a finitely generated $R$-module. Prove that $\frac{IM}{IN}$ is a finitely generated $R$-module. 2. Let $R$ be a Noetherian ring, $I$ an ideal of R. Prove that $M$ is a finitely generated $R$-module iff $\frac{M}{\text{Nil}(R)M}$ is a finitely generated $\frac{R}{\text{Nil}(R)}$-module. The first problem seems to be straightforward. Can I just take the generators and multiply them by $I$ 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 $I=\text{Nil}(R)$ for one direction; but the other direction is confusing me.

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