# Noetherian ring, finitely generated module

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• May 4th 2009, 10:19 PM
xianghu324
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.
• May 5th 2009, 12:13 AM
NonCommAlg
Quote:

Originally Posted by xianghu324
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 $i=1, \ldots, n$.

since $R$ is reduced, the nilradical of $R$ is (0) and thus $P_1 P_2 \cdots P_n=(0).$ now see my solution to part 2) of the problem in this thread: http://www.mathhelpforum.com/math-he...-r-module.html

Quote:

2. Now suppose $R$ is Artinian (need not be reduced). Prove that $M$ is Noetherian iff $M$ is Artinian.

it'd help if instead of just posting your problem, you'd also tell us what you know! for example, do you know that every Artinian ring is Noetherian? (Hopkins-Levitzki theorem) or do you know

about semisimple rings or composition series?
• May 5th 2009, 09:56 AM
xianghu324
Quote:

Originally Posted by NonCommAlg
since $R$ is reduced, the nilradical of $R$ is (0) and thus $P_1 P_2 \cdots P_n=(0).$ now see my solution to part 2) of the problem in this thread: http://www.mathhelpforum.com/math-he...-r-module.html

it'd help if instead of just posting your problem, you'd also tell us what you know! for example, do you know that every Artinian ring is Noetherian? (Hopkins-Levitzki theorem) or do you know

about semisimple rings or composition series?

Hi NonCommAlg,

We have not covered semi-simple rings yet. However, I do know that:

$M$ has a comp series $\Leftrightarrow$ $M$ is Artinian and Noetherian.

$R$ is Artin ring $\Leftrightarrow$ $R$ is Noetherian and $\text{dim} (R)=0$.

$R$ is Artin ring $\Leftrightarrow$ $R$ is Noetherian and each prime ideal is maximal.

Using comp series seems the best way to go. But I am not seeing how to use a comp series on $M$, as we have much info on M right now.
• May 5th 2009, 12:09 PM
NonCommAlg
Quote:

Originally Posted by xianghu324

2. Now suppose $R$ is Artinian (need not be reduced). Prove that $M$ is Noetherian iff $M$ is Artinian.

if $M$ is Noetherian, then it's finitely generated and we know that a finitely generated module over an Artinian ring is Artinian. conversely, suppose $M$ is Artinian. let $\overline{R}=\frac{R}{\text{Nil}(R)}.$

then $\overline{R}$ is a reduced Noetherian ring (because every Artinian ring is Noetherian). we know that $R$ has finitely many primes and every prime is maximal (so all primes are minimal).

let $P_1, \cdots , P_n$ be the prime ideals of $R.$ let $\overline{P_j}=\frac{P_j}{\text{Nil}(R)}.$ then $\overline{P_j}$ are the (minimal) primes of $\overline{R}.$ let $\overline{M}=\frac{M}{\text{Nil}(R)M}.$ then $\overline{M}$ is an $\overline{R}$ module and $\overline{M_j}=\frac{\overline{M}}{\overline{P_j} \ \overline{M}}$ is a Noetherian

$R_j=\frac{\overline{R}}{\overline{P_j}}$ module because $R_j$ is a field and $\overline{M_j}$ is an Artinian $R_j$ module and we know that over a field "Artinian" and "Noetherian" are equivalent. thus $\overline{M_j}$ is finitely generated $R_j$

module and so by part 1) of your problem, $\overline{M}$ is finitely generated $\overline{R}$ module. so by the link i already gave you, $M$ is finitely generated $R$ module and hence Noetherian, since $R$

is Noetherian.
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