Conditions under which finite length modules are artinian.

Question

We already know that finitely generated modules over artinian rings have finite length but can anyone tell me under which conditions a finitely generated module of finite length is artinian?

Thank you!

Answer 1

A module has finite length iff it is Artinian and Noetherian. This is a standard result whose proof you can find most anywhere, but I'll reproduce the particular part that you're asking about (finite length $\implies$ Artinian).

Note that a module $M$ of finite length is some finite iterated extension of its composition factors, which are simple (and thus trivially Artinian). But we know that if $$0 \to M' \to M \to M'' \to 0$$ is an exact sequence of modules, then $M$ is Artinian iff $M'$ and $M''$ are, so we win by inducting on the length of $M.$

Clearly you can replace "Artinian" with "Noetherian" everywhere to complete the $\implies$ direction of the proof.