a Noetherian , semi-Artinian module is Artinian.

Question

A semi-Artinian module is a module which every nonzero quotient of it contains a simple submodule.

How can I prove that if a module is both semi-Artinian and Noetherian, it is Artinian?

Thanks for your hints.

Answer 1

Suppose $M\ne0$ or the statement is obvious. Since $M=M/0$ has a simple submodule, the set of artinian submodules of $M$ has a maximal element $L$. If $L\ne M$, then $M/L$ has a simple submodule $S/L$. Then $$ 0\to L\to S\to S/L\to 0 $$ is an exact sequence with $L$ and $S/L$ artinian.

Answer 2

$M$ contains a simple submodule, call it $S_1$. $M/S_1$ contains a simple submodule, call it $S_2/S_1$. Continuing this way, $M/S_n$ contains a simple submodule $S_{n+1}$. During this entire process, $S_i\subseteq S_{i+1}$.

Show that $M=S_N$ for some $N$. Furthermore, the series $S_1\subseteq S_2\subseteq\ldots$ is of a special type. This will tell you that $M$ is Artinian.

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Using the same idea, it is easy to show that an Artinian semi-Noetherian module is Noetherian.