---
tag: 4C1AE84N5N
kind: concept
name: Localization
msc: [13B30]
sources:
  - url: https://en.wikipedia.org/wiki/Localization_(commutative_algebra)
same_as:
  wikidata: Q289093
  wikipedia: Localization (commutative algebra)
---

## Summary

The localization $S^{-1}R$ of a [[commutative ring|0310REJJY8]] $R$ at a [[multiplicative set|AN9SKEZ3VY]] $S$ is the ring of fractions $a/s$ with $a \in R$ and $s \in S$.
It is the most general ring in which the elements of $S$ become [[units|0XVEZDE9J8]].
It generalizes the construction of the [[field of fractions|0F2PTGX1A2]] of an [[integral domain|0QVD7WBFGN]].
Localization lets one study a ring "near" a [[prime ideal|00MHHZ9SNE]], which is the algebraic form of looking at a space near a point.

## Explanations

### L1 human

Localization means: allow some new denominators.
Starting from the integers and allowing every nonzero denominator gives the rational numbers.
Allowing only the powers of $2$ gives the fractions $a / 2^k$, such as $3/8$, but not $1/3$.
In general one picks a set $S$ of allowed denominators that contains $1$ and is closed under products.
The [[definition|KET2GSPZFG]] explains how to compare two fractions when the ring has zero divisors.