Commutative Ring Theory/Bézout domains

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 .