Nilpotent central elements annihilate simple modules ($z\in Z(A),z^r=0⟹zM=0$)

For a simple module $M$ and a central element $z$ of an algebra $A$, the image $zM$ is a submodule. If it is nonzero, it equals $M$, and iterating gives $z^{r}M=M$ for every $r$. This is impossible when $z$ is nilpotent. Hence $zM=0$. It follows, for example, that the only simple module for $k[X]/(X^r)$ is the one-dimensional quotient on which $X$ acts by zero.