# nLab localization of a ring

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.

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

### General

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.

### Localization “at” and “away from”

The common terminology in algebra is as follows.

For $S$ a set of primes, “localize at $S$” means “invert what is not divisible by $S$”; so for $p$ prime, localizing “at $p$” means considering only $p$-torsion.

Adjoining inverses $\left[{S}^{-1}\right]$ is pronounced “localized away from $S$”. Inverting a prime $p$ is localizing away from $p$, which means ignoring $p$-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 $R$ “at $p$” (or its Ab-enriched version, if saying that is necessary) really means the localization-of-the-ring R “away from p”.

## Definition

###### Definition

The localization of a ring $R$ at a multiplicative subset $S$ is the commutative ring whose underlying set is the set of equivalence classes on $R×S$ under the equivalence relation

$\left({r}_{1},{s}_{1}\right)\sim \left({r}_{2},{s}_{2}\right)\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}⇔\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\exists u\in S\phantom{\rule{thickmathspace}{0ex}}\left({r}_{1}{s}_{2}-{r}_{2}{s}_{1}\right)u=0\phantom{\rule{thickmathspace}{0ex}}\in R\phantom{\rule{thinmathspace}{0ex}}.$(r_1, s_1) \sim (r_2, s_2) \;\;\Leftrightarrow\;\; \exists u \in S \; (r_1 s_2- r_2 s_1) u = 0 \;\in R \,.

Write $r{s}^{-1}$ for the equivalence class of $\left(r,s\right)$. On this set, addition and multiplication is defined by

${r}_{1}{s}_{1}^{-1}+{r}_{2}{s}_{2}^{-1}≔\left({r}_{1}{s}_{2}+{r}_{2}{s}_{1}\right)\left({s}_{1}{s}_{2}{\right)}^{-1}$r_1 s_1^{-1} + r_2 s_2^{-1} \coloneqq (r_1 s_2 + r_2 s_1) (s_1 s_2)^{-1}
$\left({r}_{1}{s}_{1}^{-1}\right)\left({r}_{2}{s}_{2}^{-1}\right)≔{r}_{1}{r}_{2}\left({s}_{1}{s}_{2}{\right)}^{-1}\phantom{\rule{thinmathspace}{0ex}}.$(r_1 s_1^{-1})(r_2 s_2^{-1}) \coloneqq r_1 r_2 (s_1 s_2)^{-1} \,.

## 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.

Revised on November 25, 2013 01:06:12 by Urs Schreiber (89.204.137.196)