nLab
localization of a ring

Idea

Given a (possibly noncommutative) unital ring $R$ there are many situations when certain elements or matrices can be inverted in a universal way obtaining a new “localized” ring $S^{-1}R$ equipped with a localization homomorphism $R\to S^{-1}R$ under which all elements in $S$ 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 $S\subset R$ or much more generally some multiplicative set or matrices (Cohn localization) etc. There are also some specific localizations like Martindale localizations in ring theory.

Related concepts

• localization,

• localization of a commutative ring

• localization of a module

• Cohn localization, Ore localization.

References

• Zoran Škoda, Noncommutative localization in noncommutative geometry, London Math. Society Lecture Note Series 330 (pdf), ed. A. Ranicki; pp. 220–313, math.QA/0403276.