---
date: 2022-01-15 21:49
modification date: Sunday 23rd January 2022 18:54:01
title: localization
aliases: ["localization", "p-local integers", "fraction field", "local", "local ring"]
---

Last modified date: <%+ tp.file.last_modified_date() %>

---

- Tags: 
	- #CA 
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# localization of rings

# Localization

![](attachments/Pasted%20image%2020220622144606.png)

![Pasted image 20220114175002.png](attachments/Pasted%20image%2020220114175002.png)

![](attachments/Pasted%20image%2020220123193015.png)

At a prime ideal:

![](attachments/Pasted%20image%2020220123193450.png)
![](attachments/Pasted%20image%2020220123193551.png)

Localization for modules as [base change](base%20change):
![](attachments/Pasted%20image%2020220123193747.png)

When is $M$ a submodule of one of its localizations?
![](attachments/Pasted%20image%2020220123193853.png)
# Examples

[p-adic](Unsorted/p-adic.md)
![](attachments/Pasted%20image%2020220123185345.png)

Valuations for [Laurent series](Laurent%20series)
![](attachments/Pasted%20image%2020220123185457.png)

Localizations in $k[x]$, and function evaluation as a morphism onto the residue field:
![](attachments/Pasted%20image%2020220123193648.png)