---
tag: X6JD4AG973
kind: concept
name: Localization at a prime ideal
msc: [13B30, 13H99]
specializes: [4C1AE84N5N]
sources:
  - url: https://en.wikipedia.org/wiki/Localization_(commutative_algebra)
---

## Summary

The localization of a [[commutative ring|0310REJJY8]] $R$ at a [[prime ideal|00MHHZ9SNE]] $\mathfrak p$ is the ring $R_{\mathfrak p}$ of fractions whose denominators lie outside $\mathfrak p$.
It is a [[local ring|MZ02BKPHJA]] with maximal ideal $\mathfrak p R_{\mathfrak p}$.
It keeps the information about $R$ near the point $\mathfrak p$ of the [[spectrum|FB45BHHA2J]] and forgets the rest.
Many questions in commutative algebra reduce to questions about these local rings.

## Explanations

### L1 human

Pick a prime number $p$ and allow as denominators all integers not divisible by $p$.
The result $\Z_{(p)}$ contains $1/3$ and $5/7$ when $p = 2$, but not $1/2$.
In this ring every prime other than $p$ has become invertible, so only divisibility by $p$ still matters.
This is what "local at $p$" means: the ring sees only the prime $p$.
The [[definition|N37SPDWYGS]] does the same for any prime ideal of any commutative ring.