Corollary

Let $A$ be a commutative Noetherian ring, $m$ an ideal of $A$, $E$ an $A$-module and $F$ a finitely generated $A$-module. Every $A$-linear mapping $u : E \to F$ is a strict morphism (General Topology, Chapter 111, § 2, no. 8) for the $m$-adic topologies.

As $u(m^nE) = m^n u(E)$, $u$ is a strict morphism of $E$ onto $u(E)$ for the $m$-adic topologies on these two modules and the $m$-adic topology on $u(E)$ is induced by the $m$-adic topology on $F$ by Theorem 2.