---
date: 2022-03-23 14:34
modification date: Wednesday 23rd March 2022 14:34:42
title: valuation ring
aliases: [valuation ring]

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


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- Tags 
	- #NT/algebraic 
- Refs:
	- #todo/add-references
- Links:
	- [formal disk](Unsorted/formal%20disk.md)
	- [rigid geometry](rigid%20geometry.md)

---

# valuation ring

TFAE:

 - $A$ is a **valuation ring** 
 - $A$ is an integral domain $A$ with such that for every $x\in \ff(A)$, either $x\in A$ or $x\inv \in A$.
 - The ideal poset $\Id(A)$ is totally ordered.
 - The divisibility poset of $A$ is totally ordered.
 - There is a totally ordered **value group** $G$ and a [valuation](Unsorted/Valuations.md) $v: \ff(A) \to G\union\ts{\infty}$ such that 
   \[
   A = \ts{x\in \ff(A) \st v(x) \geq 0} = \ts{x\in \ff(A) \st \abs{x}_v \leq 1}
   .\]
	 - Note that this makes it look like the [ring of integers of a nonarchimedean field](Unsorted/ring%20of%20integers#Ring%20of%20integers%20of%20a%20nonarchimedean%20field) 

# Facts

- Valuation rings are [[integrally closed]].
- A [DVR](DVR.md) is an integral domain $R$ that arises as the valuation ring of $\ff(R)$ with respect to a discrete valuation. 
- The [integral closure](Unsorted/integrally%20closed.md) of an integral domain $A$ in $\ff(A)$ is the intersection of all valuation rings of $\ff(A)$ containing $A$.