Finite length $A$-algebras are finitely generated? Let $k$ be a field and $M$ a module over a (associative, unital) finite-dimensional $k$-algebra $A$.
The length of $M$ is the unique length of a composition series for $M$.
How does $M$ having finite length imply that $M$ is finitely generated as an $A$-module?
I know that there are similar statements for modules over artinian rings (and that finite-dimensional algebras are artinian rings etc.) but I can't at the moment see how to show this directly even though I've been told it's apparently easy.

A module of finite length is noetherian, so it is finitely generated.
