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 $L\prime$ are left ideals in a Gabriel filter $F$, then the set $LL\prime$ (of all products $ll\prime$ where $l\in L,l\prime \in L\prime$) 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

• Pierre Gabriel, Des catégories abéliennes
• Bo Stenstroem, Rings of quotients, Springer 1975.
• Z. Škoda, Noncommutative localization in noncommutative geometry, London Math. Society Lecture Note Series 330, ed. A. Ranicki; pp. 220–313, math.QA/0403276.