Definition 1

A commutative topological ring $A$ is said to be linearly topologized (and its topology is said to be linear) if there exists a fundamental system $\mathscr{B}$ of neighbourhoods $\mathfrak{o}$ consisting of ideals $\mathfrak{o} \subset A$.

Note that in such a ring, the ideals $3 \in \mathscr{B}$ are open and closed (General Topology, Chapter 111, § 2, no. 1, Corollary to Proposition 4 ). For all $3 \in \mathscr{B}$, the quotient topological ring $A/\mathfrak{J}$ is then discrete; for $8 \in \mathscr{B}$, $8' \in \mathscr{B}$, $\mathfrak{J}' \subset 8$, let

be the canonical mapping. We know (General Topology, Chapter III, § 7, no. 3) that $(A/\mathfrak{J}, h_{\mathfrak{J}\mathfrak{J}'})$ is an inverse system of discrete rings (relative to the indexing set $\mathscr{B}$ which is ordered by $\supset$ and directed), whose inverse limit is a complete Hausdorff linearly topologized ring $\tilde{A}$; further (loc. cit., Proposition 2 ), a strict morphism $i : A \to \tilde{A}$ is defined, whose kernel is the closure of $\{0\}$ in $A$ and whose image is everywhere dense in $\tilde{A}$, so that $\tilde{A}$ is canonically identified with the Hausdorff completion of $A$.