Let $R = k[x] / \langle x^p \rangle$, where $k$ is an (algebraically closed) field.
I want to find the finitely generated indecomposable modules over $R$ up to isomorphism, and I know that the structure theorem for finitely generated modules over a PID (somehow) tells us that in this case they will be precisely the $R$-modules: $k[x] / \langle x^l \rangle$ for $l \leq p-1$.
My question is about how to actually use the structure theorem to do this, since $R$ is not a PID in this case.
I thought about trying to project from $k[x]$ into $R$ and then to use the correspondence theorem, but I couldn't figure out how to set it up.
Can anyone help? Thanks! :)
These are just the indecomposable modules over $k[x]$ which are annihilated by $x^p$. The indecomposable modules over $k[x]$, which are finite-dimensional over $k$, are the cyclic modules $k[x]/(f^m)$ with $f$ irreducible. In this case, the only ones which count are $k[x]/(x^r)\cong R/(x^r)$ with $1\le r\le p$.