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.

Idea

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

Definition

Let $S$ be any subset of a ring $R$.

Say that a ring homomorphism $f \colon R \to A$ is $S$-inverting if the image of $S$ under $f$ is contained in the units $A^\times \subset A$, i.e. if for every $s \in S$ there is a $t \in A$ so that $t \cdot f(s) = 1 = f(s) \cdot t$ in $A$.

Then the localization of $R$ with respect to $S$ is a ring homomorphism $h \colon R \to R_S$ which is initial with respect to such $S$-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 $S$ is an Ore set, then the localization of $R$ with respect to $S$ has an explicit description in terms of fractions, see at Ore localization.

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

Examples

• The localization of a ring at a multiplicative submonoid $S$ which contains $0$ is the trivial ring.

References

See the references at Cohn localization, going back to

• Paul M. Cohn, Free rings and their relations, Academic Press (1971) [pdf]

• Paul M. Cohn, Inversive localization in noetherian rings, Communications on Pure and Applied Mathematics 26 5-6 (1973 ) 679-691 [doi:10.1002/cpa.3160260510]

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.