Let $k$ be a field and let $A$ be an $n$-dimensional commutative $k$-algebra. I have a vague recollection of the following fact, of which I am no longer sure:
$A$ has a composition series of ideals of length $n$.
Certainly, $A$ is an Artinian ring and has a composition series of $A$-submodules (=ideals) of length $\leq n$. As a $k$-module (i.e. vector space), $A$ has a composition series of $k$-submodules of length $n$ (=dimension of the $k$-vector space). Obviously, each $A$-submodule is a $k$-submodule, but not vice versa.
Is it true that one can always find among the $k$-submodules a strict chain of $A$-submodules of length $n$?
I'd appreciate a reference or a self-contained (dis)proof.