Let $A$ be a noetherian ring and $M$ an $A$-module. An element $a\in A$ is said to be locally nilpotent on $M$ if for any $x\in M$ there exists an integer $n>0$, s.t. $a^nx=0$.
Put $$p=\{a\in A\mid a \text{ is locally nilpotent on } M\}.$$ Then $p$ is clearly an ideal. Now it is stated that if $q\in \mathrm{Ass}(M)$, then there is an element $x$ of $M$ with $\mathrm{Ann}(x)=q$, and $p\subseteq q$.
I am confused here since even $a$ is locally nilpotent on $M$, it's not necessarily an annilator of $x$ (but some power is), so the last inclusion seems incorrect. Hope someone could help. Thanks!
The key thing here is that $Ann(x) = q$ is a prime ideal. If $a \in A$ is locally nilpotent then there exists some $n$ such that $a^n \in Ann(x)$, so $a \in Ann(x)$ by primality.