In Factorization, we talked about Euclidean domains and unique factorization domains. In this section, we introduce another class of integral domains called Bézout domains. As the name suggests, Bézout domains are integral domains where the Bézout identity holds. They are also a generalization of principal ideal domains.
Bézout Domain
A Bézout domain is an integral domain
such that the sum of any two principal ideals is again a principal ideal. Equivalently, for any , there exist such that .
e.g. Clearly any principal ideal domain is a Bézout domain.
Proposition
Let
be a Bézout domain. Then the followings are equivalent:
is a principal ideal domain; is a unique factorization domain; is Noetherian; satisfies the ascending chain condition on principal ideals.