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Let $M$ be a noetherian module over noetherian ring $A$.

How to prove that there exists submodule $N\subset M$ such that $$M/N\cong A/\mathfrak{p}$$ for some prime ideal $\mathfrak{p}\in A$.

Is it true that any submodule of noetherian module over noetherian ring is noetherian? (because it is finitely generated as submodule of noetherian module?)

Thanks a lot!

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    can you see what happens if you take M' maximal?2012-10-12
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    @MarianoSuárez-Alvarez: any maximal? any "limit" of ascending chain?2012-10-12
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    In fact the ideal $\mathfrak{p}$ can be taken maximal: choose $N$ maximal in the set of proper submodules of $M$ (why there is such $N$?); then $M/N$ is a simple module, so...2012-10-12

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