---
tag: HGEX40P0Z0
kind: concept
name: Bézout domain
msc: [13F05, 13G05]
specializes: [ACDEK1GD9G]
sources:
  - url: https://en.wikipedia.org/wiki/B%C3%A9zout_domain
same_as:
  wikidata: Q2386260
  wikipedia: Bézout domain
---

## Summary

A Bézout domain is an [[integral domain|09S3PFBEPM]] in which every finitely generated ideal is [[principal|00A2EDFT9Q]].
In such a ring [[Bézout's identity|0DQVHMYAGA]] holds: a gcd of $a$ and $b$ can be written as $ax + by$.
Every [[principal ideal domain|0RZYDVZ475]] is a Bézout domain, and a Noetherian Bézout domain is a principal ideal domain.
Bézout domains are [[GCD domains|ACDEK1GD9G]] and [[Prüfer domains|2RDAA4PCK8]].

## Explanations

### L1 human

In the integers, $\gcd(a, b)$ can always be written as $ax + by$ for some integers $x, y$.
A Bézout domain is a ring where this works for every pair of elements, even if the ring has ideals that are not finitely generated.
The ring of entire functions on $\C$ and the ring of all algebraic integers are Bézout domains that are not principal ideal domains.
The polynomial ring $\Z[x]$ is not Bézout: the ideal generated by $2$ and $x$ is not principal.