  In this article we study two classes of integral domains. The first is
characterized by having a finite intersection of principal ideals being
finitely generated only when it is principal. The second class consists of the
integral domains in which a finite intersection of principal ideals is always
non-finitely generated except in the case of containment of one of the
principal ideals in all the others. We relate these classes to many
well-studied classes of integral domains, to star operations and to classical
and new ring constructions.
