# Problem of the Week #235 - Nov 29, 2016

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#### Euge

##### MHB Global Moderator
Staff member
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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#### Euge

##### MHB Global Moderator
Staff member
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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