Exercise 11, § 7

Let A be a noetherian ring, and let a, b be two ideals of A such that Supp(a) = Supp(b) = Spec(A). Now prove that the following conditions are equivalent : (i) There exists an a-module of type finite M such that Supp(M) = Spec(A) and such that b.M $\subset a.M$. (ii) We have $b \subset \overline{a}$.