Exercise 16, § 2

Let $A$ be a topological ring (not necessarily commutative). The topology on $A$ is called (left) linear if the open left ideals of $A$ form a fundamental system of neighbourhoods of 0. If this is so, the set $\mathscr{F}$ of open left ideals of $A$ satisfies the following conditions: (1) Every left ideal of $A$ containing an ideal $m \in \mathscr{F}$ belongs to $\mathscr{F}$. (2) Every finite intersection of ideals of $\mathscr{F}$ belongs to $\mathscr{G}$. (3) If $m \in \mathscr{F}$ and $a \in A$, the left ideal $n$ consists of those $x \in A$ such that $xa \in m$ belongs to $\mathscr{F}$.

Conversely, if a set $\mathscr{F}$ of left ideals of a ring $A$ satisfies these three conditions, there exists a unique topology compatible with the ring structure on $A$ and for which $\mathscr{F}$ is the set of open left ideals of $A$. Such a set $\mathscr{F}$ of left ideals is called (left) topologizing.