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Let $R$ be a commutative ring with identity, $M$ an $R$-module and $N$ is a submodule of $M$. Which of the following statements true?

  1. If every submodule of $M$ is Artinian, then $M$ is Artinian.

  2. If $M/N$ is Artinian, then $M$ is Artinian.

  3. If $N$ is Artinian, then $M/N$ is Artinian.

Clearly (1) is true. (2) is false because consider the $\mathbb{Z}$-module $\mathbb{Q}$. But (3)?

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up vote 2 down vote accepted

For (3): What if you take $N = 0$ and $M = \Bbb{Z}$?

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