and in general context in his book with Zisman. By Gabriel localization one usually means a specific class of localizations of rings and the corresponding localization of categories of modules over rings.
Given a (possibly noncommutative and nonunital) ring and a Gabriel filter of left ideals in , a Gabriel localization endofunctor
is defined in one of the number of equivalent ways.
For example, for any uniform filter of left ideals in one defines a subfunctor of the identity functor on the category of left -modules
In a later work of Goldman was called a radical functor. If is not only uniform but in fact a Gabriel filter then the radical is idempotent, i.e. . If is unital, is equivalent to the functor given on objects by
For each uniform fiter one also defines the endofunctor on by
(the colimit is over downward directed family of ;eft ideals in and is a colimit of a functor with values in the category of abelian groups; the uniformness condition however gurantees that there is a canonical structure of an -module on the colimit group ).
Finally, for the Gabriel filter one defines the Gabriel (endo)functor on objects by
The essential image of the functor is the localized category. The left -module has a canonical structure of a ring over ; there is a natural forgetful functor from the localized category to the category of left -modules. Under strong assumptions on the filter this functor is in fact an equivalence of categories, e.g. when the localization is Ore.
Zoran Škoda, Localizations for construction of quantum coset spaces, in “Noncommutative geometry and Quantum groups”, W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265–298, Warszawa 2003, math.QA/0301090.