Exercise 8, § 2

Let $A$ be a Noetherian ring, $E$ a finitely generated $A$-module, $F$ a submodule of $E$ and $m$ an ideal of $A$. Let $F = \bigcap_{p \in \operatorname{Ass}(E/F)} Q(p)$ be a reduced primary decomposition of $F$ in $E$. Show that the closure of $F$ in $E$ under the $m$-adic topology on $E$ is equal to $\bigcap_{p \in \Phi} Q(p)$, where $\Phi$ is the set of $p \in \operatorname{Ass}(E/F)$ such that $p + m \neq A$. (Consider first the case where $F$ is $p$-primary in $E$ and show that, if $p + m = A$, $F$ is dense in $E$; proceed to the general case using Exercise 15 of § 1 and Chapter 111, § 3, no. 4, Corollary 1 to Theorem 3).