nLab localization of a ring


This entry is meant to be about the general notion of localization of a possible noncommutative ring (a special case of Cohn localization). For the more restrictive but traditional notion for commutative rings see at localization of a commutative ring.



Given a (possibly noncommutative) unital ring RR one may ask to universally force some subset SS of its elements to become multiplicatively invertible in that there is a “localized” ring S 1RS^{-1}R equipped with a universal localization homomorphism RS 1RR\to S^{-1}R under which all elements in SS are mapped to units.

This is a special case of and generalizes to the notion of Cohn localization where one may also force certain matrices with coefficients in the ring to become invertible.

Often one inverts elements in a left or right Ore subset SRS\subset R in which case the localized ring is expressed by fractions as naively expected, in which case one speaks of Ore localization.

This Ore condition is automatic for commutative rings which leads to the notion of localization of a commutative ring.

Another special case is known as Martindale localization.


Let SS be any subset of a ring RR.

Say that a ring homomorphism f:RAf \colon R \to A is SS-inverting if the image of SS under ff is contained in the units A ×AA^\times \subset A, i.e. if for every sSs \in S there is a tAt \in A so that tf(s)=1=f(s)tt \cdot f(s) = 1 = f(s) \cdot t in AA.

Then the localization of RR with respect to SS is a ring homomorphism h:RR Sh \colon R \to R_S which is initial with respect to such SS-inverting ring homomorphisms.

By this defining universal property the localization is unique up to isomorphism, when it exists. Its existence is a special case of universal localization/Cohn localization, a general abstract construction.

If SS is an Ore set, then the localization of RR with respect to SS has an explicit description in terms of fractions, see at Ore localization.

If SS is a submonoid of the center Z(R)Z(R) of the multiplicative monoid of RR, then the localization of RR at SS follows the same definition as that of localization of a commutative ring.


  • The localization of a ring at a multiplicative submonoid SS which contains 00 is the trivial ring.


See the references at Cohn localization, going back to

with more discussion in:

  • V. Retakh, R. Wilson, Advanced course on quasideterminants and universal localization (2007) [pdf]

  • Andrew Ranicki (ed.), Noncommutative localization in algebra and topology, (Proceedings of Conference at ICMS, Edinburgh, 29-30 April 2002) London Math. Soc. Lecture Notes Series 330 Cambridge University Press (2006) [pdf]

and see also at noncommutative localization.

For references on the localization of commutative rings see there.

See also:

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

Last revised on February 9, 2023 at 22:57:53. See the history of this page for a list of all contributions to it.