nLab Gabriel filter


A Gabriel filter GG is a uniform filter of left ideals in a ring RR 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 00 for a topology on RR; on the other hand the Gabriel filter itself is an additive analogue of Grothendieck topology, see also enriched sheaf.


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


A version for quantales:

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

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