Simple modules over formal power series in one variable.

Question

What are the simple finite dimensional modules over $k[[x]]$ (the ring of formal power series over a field $k$)?

Since $k[[x]]$ is a principal ideal domain, such modules much be torsion, but i can't figure out anything more. Thank you.

Answer 1

Let $M$ be a simple module over $k[[x]]$. Let $m\in M$, $m\neq 0$ be arbitrary. Then $k[[x]]\rightarrow M$, $f(x)\mapsto f(x).m$ is surjective, since $M$ is simple and the image is a non-zero submodule. Denote the kernel by $I$, so that we have an isomorphism $k[[x]]/I\cong M$ of $k[[x]]$-modules. Since $M$ is simple, so is $k[[x]]/I$, whose submodules correspond to submodules of $k[[x]]$ containing $I$. This shows that $I$ is a maximal submodule/ideal of $k[[x]]$.
As $k[[x]]$ is a local ring with maximal ideal $(x)$, it follows that $I = (x)$ and hence $M = k[[x]]/(x) = k$. Therefore, $k$ is the only simple $k[[x]]$-module, where $x$ acts by zero.

Note that this argument is valid for any field $k$.