uniform filter

Given a ring RR, for any left ideal IRI\subset R and a set SRS\subset R define

(I:S)={rR|rSI}. (I : S) = \{ r \in R \,|\, r S \subset I \}.

This is clearly a left ideal again. The special case (I:R)(I:R) is a two-sided ideal, namely the maximal ideal of RR contained in II. If rRr\in R then we write (I:r):=(I:{r})(I:r) := (I:\{r\}).

A filter FF in the lattice of left ideals of a ring RR is a uniform filter if IFI\in F implies (I:r)F(I:r)\in F for any rRr\in R. Equivalently, the Gabriel composition of filters satisfies FF{R}F\subset F\bullet \{R\}. The Gabriel composition of uniform filters is a uniform filter. Uniform filters are also called topologizing, because a non-empty set of left ideals of RR is a uniform filter iff it is the family of left ideals of RR which form an open neighborhood of 00 in a “linear topology” on RR.

The uniform filters of ideals in a ring RR bijectively correspond to kernel functors on RR-Mod\mathrm{Mod} (left exact subfunctors of the identity functor). The correspondence goes as follows. If FF is a uniform filter, and MM in RR-Mod\mathrm{Mod}, define σ FM\sigma_F M as the set of all mMm\in M such that mm is annihilated by some left ideal II in FF. Conversely, given a kernel functor σ\sigma, define a uniform filter F σF^\sigma to be the filter whose members are all left ideals II such that σ(R/I)=R/I\sigma(R/I)=R/I.

The most important class of uniform filters are Gabriel filters.

category: algebra