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.