Exercise 7, § 3

Let $A$ be a commutative Noetherian ring and $E$ an $A$-module. Let $A$ be given the topology $\mathscr{T}_m(A)$ (§ 2, Exercise 24) and let $\mathscr{T}_m(E)$ denote the linear topology on $E$ for which a fundamental system of neighbourhoods of 0 consists of the submodules $a.E$, where $a$ runs through a fundamental system of neighbourhoods of 0 for $\mathscr{T}_m(A)$, consisting of ideals of $A$. Show that if $E$ is finitely generated, $E$ is a Hausdorff topological $A$-module (with $\mathscr{T}_m(A)$ and $\mathscr{T}_m(E)$) where every submodule is closed. A fundamental system of neighbourhoods of 0 for $\mathscr{T}_m(E)$ then consists of the submodules $F$ of $E$ such that $E/F$ is a module of finite length and, for every submodule M of E, the topology $\mathscr{T}_m(M)$ is induced by $\mathscr{T}_m(E)$.