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 $\operatorname{gcd}(a,b)$ denotes one such greatest common divisor between $a,b\in D$ , then for some $r,s\in D$ :

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

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