The noncommutative localization is a common term for the localization in noncommutative algebra, that is the localizations of noncommutative algebraic structures, most often noncommutative rings and associative algebras, as well as localization functors on various categories of modules (and bimodules) over possibly noncommutative algebras. Most often this term is used for several kinds of localization which occur in noncommutative ring theory and in particular the (left or right) Ore localization, Gabriel localization and Cohn universal localization of rings and of their associated categories of modules. Many notion here have straightforward extension to general Grothendieck categories.

Gabriel’s locazalization is usually stated in terms of Gabriel filters, but it fits into a more general abelian localization of abelian categories? on thick subcategories.

- Pierre Gabriel, Des catégories abéliennes, Bulletin de la Société Mathématique de France, 90 (1962), p. 323-448, numdam

An alternative, but equivalent, approach to Gabriel localization, via kernel functors is introduced in

The language of torsion theories was originally developed also in this context, but in fact it can pertain to a more general situation than to the categories of modules over rings. Hereditary torsion theories on the categories of 1-sided modules correspond to Gabriel lcoalization and the localization functors are flat. Cohn universal localizations of rings corresponds at the level of categories of modules to certain nonhereditary torsion theories.

See also torsion theory, localization, Q-category, thick subcategory, topologizing subcategory, Gabriel filter

- Bo Stenström,
*Rings and modules of quotients*. Lecture Notes in Mathematics**237**, Springer-Verlag 1971. vii+136 pp. MR0325663 - Nicolae Popescu,
*An introduction to Abelian categories with applications to rings and modules*, Acad. Press 1973 - A. W. Goldie,
*Torsion-free modules and rings*, J. Algebra**1**1964, 268–287 MR164991 doi - Zoran Škoda,
*Noncommutative localization in noncommutative geometry*, London Math. Society Lecture Note Series**330**, ed. A. Ranicki; pp. 220–313, math.QA/0403276

category: algebraic geometry

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