Nilpotent central elements annihilate simple modules
= Nilpotent central elements annihilate simple modules
{title2=$z\in Z(A),\ z^r=0\Longrightarrow zM=0$}
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 $z^rM=M$ 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.