This entry is about the general notion of localization of a possible noncommutative ring. For the more restrictive but more traditional notion of localization of a commutative ring see there.
symmetric monoidal (∞,1)-category of spectra
Given a (possibly noncommutative) unital ring there are many situations when certain elements or matrices can be inverted in a universal way obtaining a new “localized” ring equipped with a localization homomorphism under which all elements in are mapped to multiplicatively invertible elements (units). The latter property must be modified for Cohn localization at multiplicative set of matrices.
We can typically invert elements in a left or right Ore subset or much more generally some multiplicative set or matrices (Cohn localization) etc. There are also some specific localizations like Martindale localizations in ring theory.
The common terminology in algebra is as follows.
Adjoining inverses is pronounced “localized away from ”. Inverting a prime is localizing away from , which means ignoring -torsion.
Evidently, this conflicts with more-categorial uses of “localized”; “inverting weak equivalences” is called localization, by obvious analogy, and is written as “localizing at weak equivalences”. This is confusing! It’s also weird: since a ring is a one-object Ab-enriched category with morphisms “multiply-by”, the localization-of-the-category “at ” (or its Ab-enriched version, if saying that is necessary) really means the localization-of-the-ring R “away from p”.
Write for the equivalence class of . On this set, addition and multiplication is defined by
(e.g. Stacks Project, def. 10.9.1)
Basics of localizations of commutative rings are reviewed for instance in
Discusison of the general concept in noncommutative geometry is in