Example for a non-semisimple algebra

Question

Why is the $\mathbb{C}[X]/\langle X^2\rangle$-module $\mathbb{C^2}$ with $X$ acting as $\begin{pmatrix}0&1\\0&0\end{pmatrix}$ not a direct sum of simple modules?

As far as I understand, the action is $[X]\cdot(x,y)=(y,0)$.

Answer 1

Simple modules over a commutative ring $R$ are of the form $R/\mathfrak{m}$ for a maximal ideal $\mathfrak{m}$. The only maximal ideal of $\Bbb C[X]/(X^2)$ is $(X)/(X^2)$, so the only simple module is $(\Bbb C[X]/(X^2))/((X)/(X^2)) \cong \Bbb C$ with $X$ acting trivially. Thus $X$ acts trivially on any direct sum of simple modules.