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Consider the short exact sequence of modules

$$0\rightarrow M\rightarrow N\rightarrow K\rightarrow0$$ Let $K$ be an Artinian (Noetherian) module, can we get $N$ is an Artinian (Noetherian) module?

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    $\begingroup$ $K$ could be the zero module, and $M \cong N$, so it seems you'll need more assumptions than this. $\endgroup$
    – mollyerin
    Jan 13, 2015 at 22:31

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The result, as concerns short exact sequences, is that $N$ is artinian if and only if $M$ and $K$ are artinian.

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  • $\begingroup$ Thank you for your answer Bernard. I knew that, now I wanted to see have been thinking on this subject with this condition until now? $\endgroup$
    – Sara
    Jan 16, 2015 at 17:46
  • $\begingroup$ @Sara: I don't quite understand what your question is? $\endgroup$
    – Bernard
    Jan 17, 2015 at 14:10

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