Semi-simple $\mathbb R[x]$-module structure on $\mathbb R^2$ This might just be a trivial question:

Let $R=\mathbb{R}[x]$ and define an $R$-module structure on $\mathbb{R}^2$ with $x$ acting by $\begin{pmatrix}0&1\\-1&0\end{pmatrix}$. Is the module semi-simple?

So my thought is: since the matrix admits no eigenvalue in $\mathbb{R}$, we cannot find any non-trivial submodule, thus this is a simple module, thus semi-simple. Is this a correct understanding? What's the correct/better way to understand this?

Let $A=\begin{pmatrix}0&1\\-1&0\end{pmatrix}$. The characteristic polynomial of $A$ is $x^2+1$, and thus the module is isomorphic to $\mathbb R[x]/(x^2+1)$ (since $x^2+1$ is irreducible over $\mathbb R$), hence it is simple.
