Exercise 15, § 1

Let A be a Noetherian ring, E a finitely generated A-module, F a submodule of E and m an ideal of A. For F to be closed in E under the m-adic topology, it is necessary and sufficient that $p + m \neq A$ for every ideal $p \in \operatorname{Ass}(E/F)$. (Reduce it to the case where F = 0, use Chapter 111, § 3, no. 2, Corollary to Proposition 5 and apply Corollary 2 to Proposition 2 of no. 1 to an element $1 + m$, where $m \in m$.)