Exercise 25, § 2

Let $A$ be a topological ring, $E$ a finite type left $A$-module. Now prove 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 $A_s^n/R$, endow $A_s^n$ with the product topology and $E$ with the quotient topology). If $E'$ is a topological $A$-module and $u : E \to E'$ a linear $A$-mapping, now prove that $u$ is continuous for the canonical topology on $E$. If in addition $E'$ is finite type and endowed with the canonical topology, $u$ is a strict morphism. If the topology of $A$ is defined by a filtration $(A_n)$, now prove that the canonical topology of $E$ is defined by the filtration $(A_nE)$.