Corollary

Let $A$ be a Noetherian ring. $M$ an $A$-module and $a$ an element of $A$. For the homomorphism $a_M : x \mapsto ax$ of $M$ to be almost nilpotent, it is necessary and sufficient that $a$ belong to every ideal of $\operatorname{Ass}(M)$.

The condition of the statement is equivalent to saying that for all $x \in M$ there exists $n(x) > 0$ such that $(Aa)^{n(x)}(Ax) = 0$; by Proposition 9 this means also that $a$ belongs to all the prime ideals associated with the submodule $Ax$ of $M$; the corollary then follows from the fact that $\operatorname{Ass}(M)$ is the union of the $\operatorname{Ass}(Ax)$ where $x$ runs through $M$ (no. 1, formula (1)).