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

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 neighborhoods of $0$ for a topology on $R$; on the other hand the Gabriel filter itself is an additive analogue of Grothendieck topology, see also enriched sheaf.

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.

- Pierre Gabriel,
*Des catégories abéliennes* - Bo Stenstroem?,
*Rings of quotients*, Springer 1975. - Zoran Škoda,
*Noncommutative localization in noncommutative geometry*, London Math. Society Lecture Note Series**330**, ed. A. Ranicki; pp. 220–313, math.QA/0403276. - Wendy Lowen,
*A generalization of the Gabriel–Popescu theorem*, J. Pure Appl. Alg.**190**:1–3 (2004) 197-211 doi

A version for quantales:

- Kimmo I. Rosenthal,
*A general approach to Gabriel filters on quantales*, Comm. Algebra**20**, n.11 (1992) 3393–3409 doi

Last revised on October 11, 2023 at 19:49:20. See the history of this page for a list of all contributions to it.