Bezout domain

A Bezout domain D is an integral domainMathworldPlanetmath such that every finitely generatedMathworldPlanetmathPlanetmath ideal of D is principal (


  • A PID is obviously a Bezout domain.

  • Furthermore, a Bezout domain is a gcd domain. To see this, suppose D is a Bezout domain with a,bD. By definition, there is a dD such that (d)=(a,b), the ideal generated byPlanetmathPlanetmath a and b. So a(d) and b(d) and therefore, da and db. Next, suppose cD and that ca and cb. Then both a,b(c) and so d(c). This means that cd and we are done.

  • From the discussion above, we see in a Bezout domain D, a greatest common divisorMathworldPlanetmath exists for every pair of elements. Furthermore, if gcd(a,b) denotes one such greatest common divisor between a,bD, then for some r,sD:


    The above equation is known as the Bezout identity, or Bezout’s Lemma.

Title Bezout domain
Canonical name BezoutDomain
Date of creation 2013-03-22 14:19:53
Last modified on 2013-03-22 14:19:53
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 10
Author CWoo (3771)
Entry type Definition
Classification msc 13G05
Synonym Bézout domain
Related topic GcdDomain
Related topic DivisibilityByProduct
Defines Bezout identity