linear topological ring

Linear topological rings

Linear topological rings


A topological ring RR 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 RR is a left uniform filter (=topologizing filter) of left ideals iff it is a basis of neighborhood of 00 of some left linear topology. In other words, the following conditions hold:

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

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

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

Uniform filter condition:

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

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

is open.

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



Last revised on November 28, 2020 at 22:54:00. See the history of this page for a list of all contributions to it.