Why is $M/(N\cap P)$ Artinian when $M/N$ and $M/P$ are Artinian?

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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 http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

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No one answered this week's problem. You can read my solution below.

Spoiler

$M/(N\cap P)$ 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.