Definition (Bézout domain):
A Bézout domain is an integral domain whose every finitely generated ideal is principal, ie. generated by a single element.
Proposition (Every Bézout domain is a GCD domain):
Let
be a Bézout domain. Then
is a GCD domain.
Proof: Given any two elements
, we may consider the ideal
generated by
and
. By the definition of Bézout domains,
for at least one
(which is moreover unique up to similarity). Then
and
, so that by the characterisation of divisibility by principal ideals,
is a common divisor of
and
. Moreover, if
is another common divisor of
and
, then
, so that
, so that
. Hence,
is a greatest common divisor of
and
.