nLab linear topological ring

Idea

A topological ring $R$ is said to have a (left) linear topology if it has a topological base of neighbourhoods of 0 consisting of (left) ideals.

Taking this apart we have a definition/proposition of such a topology, given by a left uniform filter:

Proposition (Gabriel)

A nonempty family of left ideals in $R$ is a left uniform filter (=topologizing filter) of left ideals iff it is a basis of neighborhood of $0$ of some left linear topology. In other words, the following conditions hold:

Filter axioms: (0) $\{R\}$ is open and:

(i) If $\mathfrak{a}\subseteq \mathfrak{b}$ are left ideals and $\mathfrak{a}$ is open, then so is $\mathfrak{b}$.

(ii) If $\mathfrak{a}$ and $\mathfrak{b}$ are open left ideas, then so is $\mathfrak{a}\cap\mathfrak{b}$.

Uniform filter condition:

(UF) If $\mathfrak{a}$ is an open left ideal and $r\in R$, then the left ideal

$$(\mathfrak{a}:r)= \{x\in R\mid x r\in \mathfrak{a}\}$$

is open.

Note that if $R$ is commutative then (i) implies (UF).

Examples

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References

• related ideas include pseudocompact ring, Gabriel filter, topologizing subcategory, Gabriel-Oberst duality, linearly compact module

• Pierre Gabriel, Des catégories abéliennes, Bulletin de la Société Mathématique de France, 90 (1962), p. 323-448 (numdam)

• U. Oberst, Duality theory for Grothendieck categories and linearly compact rings, J. Algebra 15 (1970), p. 473 –542, pdf