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.