Proposition 8

Let A be a Noetherian ring, a an ideal of A and M a finitely generated A-module. The following conditions are equivalent:

(a) there exists an element x $\# O$ of M such that $ax = 0$. (b) for all $a \in a$, there exists an element x $\neq O$ of M such that $ax = 0$; (c) there exists $p \in \operatorname{Ass}(M)$ such that $a \subset p$.

Clearly (a) implies (b). By virtue of no. 1, Corollary 2 to Proposition 2, condition (b) means that the ideal a is contained in the union of the prime ideals associated with M and hence in one of them since Ass(M) is finite (Chapter 11, § 1, no. 1, Proposition 2); thus (b) implies (c). Finally, if there exists $p \in \operatorname{Ass}(M)$ such that $a \subset p$, p is the annihilator of an element $x \neq 0$ of M (no. 1, Definition 1) and $ax = 0$; thus (c) implies (a).