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Math Help - Noetherian ring, finitely generated module

  1. #1
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    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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  2. #2
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    Quote Originally Posted by xianghu324 View Post
    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


    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?
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    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.
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  4. #4
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    Quote Originally Posted by xianghu324 View Post

    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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