# nLab Gabriel filter

## Definition

A Gabriel filter $G$ is a uniform filter of left ideals in a ring $R$ which is idempotent under Gabriel composition of filters.

## Terminology

In our definition this notion is equivalent to topologizing filter; though for some authors the latter notion slightly differs. Stenstroem says Gabriel topology instead of Gabriel filter, because all Gabriel filters form a basis of nieghborhoods of $0$ for a topology on $R$.

## Properties

If $L$ and $L'$ are left ideals in a Gabriel filter $F$, then the set $L L'$ (of all products $l l'$ where $l\in L, l'\in L'$) is an element on $F$. Any uniform filter $F$ is contained in a minimal Gabriel filter $G$ (said to be generated by $F$), namely the intersection of all Gabriel filters containing $F$. Given a Gabriel filter $G$, the class of all $G$-torsion modules (see uniform filter) is a hereditary torsion class.

## Literature

Revised on December 14, 2010 19:59:50 by Zoran Škoda (161.53.130.104)