Let $P\in \mathbb{F}[x]$ be a monic polynomial of degree $n$ over a field $\mathbb{F}$, and $M_P$ its companion matrix. The matrix $M_P$ gives a module of $\mathbb{F}[x]$ on $\mathbb{F}^n$, by letting $x\cdot w= M_P\cdot w$ for $w \in \mathbb{F}^n$.
This $\mathbb{F}[x]$-module is isomorphic to $\mathbb{F}[x]/(P)$ and therefore cyclic, true? It is simple iff $P$ is irreducible, right? When is it indecomposable? When is it semisimple?