Exercise 14, § 2

Let $A$ be a ring (not necessarily commutative) and $E$ a left $A$-module. A topology on $E$ is called linear if it is invariant under translations and $0$ admits a fundamental system of neighbourhoods which are submodules of $E$; $E$ is then said to be linearly topologized. A linear topology on $E$ is compatible with its additive group structure and defines on $E$ a topological $A$-module structure if $A$ is given the discrete topology. On any $A$-module the discrete topology and the coarsest topology are linear topologies.

(a) If $E$ is a linearly topologized module and $F$ a submodule of $E$, the induce topology on $F$ and the quotient topology of that on $E$ by $F$ are linear topologies. If $(E_\alpha, f_{\alpha\beta})$ is an inverse system of linearly topologized $A$-modules, where the $f_{\alpha\beta}$ are continuous linear mappings, the topological $A$-module $E = \lim \leftarrow E_\alpha$ is linearly topologized.

(b) Let $E$ be a linearly topologized $A$-module. There exists a fundamental system $(V_\lambda)_{\lambda \in L}$ of neighbourhoods of $0$ in $E$ consisting of open (and closed) submodules; if $\phi_{\lambda\mu}: E/V_\mu \to E/V_\lambda$ is the canonical mapping when $V_\lambda \supseteq V_\mu$, the family $(E/V_\lambda, \phi_{\lambda\mu})$ is an inverse system of discrete $A$-modules; the topological $A$-module $\tilde{E} = \lim \leftarrow E/V_\lambda$ is identified with the Hausdorff completion of $E$, which is therefore linearly topologized (General Topology, Chapter III, § 7, no. 3).

(c) A Hausdorff linearly topologized A-module $E$ is discrete if it is Artinian or if there exists a least element in the set of submodules $\neq 0$ of $E$.