Exercise 14, § 2

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

a) If $E$ is a linearly topologized module, $F$ a submodule of $E$, the topology induced on $F$ and the quotient topology of that of $E$ by $F$ are linear topologies. If $(E_\alpha, f_{\alpha\beta})$ is a projective 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 $E$, formed of open (and closed) submodules; if $\varphi_{\lambda\mu} : E/V_\mu \to E/V_\lambda$ is the canonical mapping when $V_\lambda \supset V_\mu$, the family $(E/V_\lambda, \varphi_{\lambda\mu})$ is a projective system of discrete $A$-modules; the topological $A$-module $\tilde{E} = \lim \leftarrow E/V_\lambda$ identifies with the separated completion of $E$, which is therefore linearly topologized (Top. gén., chap. III, 3e éd., § 7, no 3).

c) A separated linearly topologized $A$-module $E$ is discrete when it is artinian, or when there exists a smallest element in the set of submodules $\neq 0$ of $E$.