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Bezout domain
A Bezout domain $D$ is an integral domain such that every finitely generated ideal of $D$ is principal.
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 $\operatorname{gcd}(a,b)$ 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.
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