nLab uniform filter

Idea

Given a ring $R$, for any left ideal $I\subset R$ and a set $S\subset R$ define

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

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

A filter $F$ in the lattice of left ideals of a ring $R$ is a uniform filter if $I\in F$ implies $(I:r)\in F$ for any $r\in R$. Equivalently, the Gabriel composition of filters satisfies $F\subset F\bullet \{R\}$. 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 $R$ is a uniform filter iff it is the family of left ideals of $R$ which form an open neighborhood of $0$ in a “linear topology” on $R$.

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

The most important class of uniform filters are Gabriel filters.

References

• Garcia Román, Manuel; Márquez Hernández, Mercedes; Jara, Pascual; Verschoren, Alain, Uniform filters, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 40 no. 2 (1999), p. 82-126 numdam

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

category: algebra