Proposition 9

Let A be a Noetherian ring, a an ideal of A and M a finitely generated A-module. For there to exist an integer $n > 0$ such that $a^n M = 0$, it is necessary and sufficient that a be contained in the intersection of the prime ideals associated with M.

This intersection is also that of the minimal elements of Supp(M) (no. 3, Corollary 1 to Proposition 7) and to say that a is contained in this intersection is equivalent to saying that $V(a) \supset \operatorname{Supp}(M)$ in the notation of Chapter II, § 4; the conclusion then follows from Chapter 11, § 4, no. 4, Corollary 2 to Proposition 17.