Exercise 25, § 2

Let $A$ be a topological ring and $E$ a finitely generated left $A$-module. Show that there exists on $E$ a topology (called canonical) compatible with its $A$-module structure and finer than all the others (write $E$ in the form $A_s^n/R$, give $A_s^n$ the product topology and $E$ the quotient topology). If $E'$ is a topological $A$-module and $u : E \to E'$ an $A$-linear mapping, show that $u$ is continuous with the canonical topology on $E$. If further $E'$ is finitely generated and has the canonical topology, $u$ is a strict morphism. If the topology on $A$ is defined by a filtration $(A_n)$, show that the canonical topology on $E$ is defined by the filtration $(A_n E)$.