Gabriel filter

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 nieghborhoods of $0$ for a topology on $R$.

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. - Z. Škoda,
*Noncommutative localization in noncommutative geometry*, London Math. Society Lecture Note Series**330**, ed. A. Ranicki; pp. 220–313, math.QA/0403276.