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.

• 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 $a,b\in D$. 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.

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

  $$\gcd (a,b)=ra+sb.$$

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

Title	Bezout domain
Canonical name	BezoutDomain
Date of creation	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