A topological ring 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:
A nonempty family of left ideals in is a left uniform filter (=topologizing filter) of left ideals iff it is a basis of neighborhood of of some left linear topology. In other words, the following conditions hold:
Filter axioms: (0) is open and:
(i) If are left ideals and is open, then so is .
(ii) If and are open left ideas, then so is .
Uniform filter condition:
(UF) If is an open left ideal and , then the left ideal
is open.
Note that if is commutative then (i) implies (UF).
…
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
Last revised on November 29, 2020 at 03:54:00. See the history of this page for a list of all contributions to it.