Bezout domain
A Bezout domain $D$ is an integral domain^{} such that every finitely generated^{} ideal of $D$ is principal (http://planetmath.org/PID).
Remarks.

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A PID is obviously a Bezout domain.

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Furthermore, a Bezout domain is a gcd domain. To see this, suppose $D$ is a Bezout domain with $a,b\in D$. By definition, there is a $d\in D$ such that $(d)=(a,b)$, the ideal generated by^{} $a$ and $b$. So $a\in (d)$ and $b\in (d)$ and therefore, $d\mid a$ and $d\mid b$. Next, suppose $c\in D$ and that $c\mid a$ and $c\mid b$. Then both $a,b\in (c)$ and so $d\in (c)$. This means that $c\mid d$ and we are done.

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From the discussion above, we see in a Bezout domain $D$, a greatest common divisor^{} exists for every pair of elements. Furthermore, if $\mathrm{gcd}(a,b)$ denotes one such greatest common divisor between $a,b\in D$, then for some $r,s\in D$:
$$\mathrm{gcd}(a,b)=ra+sb.$$ The above equation is known as the Bezout identity, or Bezout’s Lemma.
Title  Bezout domain 

Canonical name  BezoutDomain 
Date of creation  20130322 14:19:53 
Last modified on  20130322 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 