Proposition 1

Let $A$ be a Noetherian ring and $M$ an $A$-module. The following conditions are equivalent: (a) $\operatorname{Ass}(M)$ is reduced to a single element.

(b) $M \neq 0$ and every homothety of $M$ is either injective or almost nilpotent ($§ 1$, no. 4). If these conditions are fulfilled and $p$ is the set of $a \in A$ such that the homothety $a_M$ is almost nilpotent, then $\operatorname{Ass}(M) = \{p\}$.

This follows immediately from $§ 1$, no. 4, Corollary to Proposition 9 and no. 1, Corollary 2 to Proposition 2.