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Problem of the Week #235 - Nov 29, 2016

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Euge

MHB Global Moderator
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Jun 20, 2014
1,896
Here is this week's POTW:

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Let $R$ be a commutative ring. If $N$ and $P$ are submodules of an $R$-module $M$ such that $M/N$ and $M/P$ are Artinian, show that $M/(N\cap P)$ is Artinian.

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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Euge

MHB Global Moderator
Staff member
Jun 20, 2014
1,896
No one answered this week's problem. You can read my solution below.



$M/(N\cap P)$ is isomorphic to a submodule of the Artinian module $M/N \times M/P$ via the $R$-mapping $M/(N\cap P) \to M/N \times M/P$ given by $m + N\cap P \mapsto (m + N, m + P)$; hence, $M/(N\cap P)$ is Artinian.
 
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