A characterization of Krull domains in terms of their factor rings

Abstract

ABSTRACT Let D be an integral domain, w be the so-called w-operation on D, and be the set of nonzero finitely generated ideals J of D with . Let I be a w-ideal of D, D/I be the factor ring of D modulo I, be the injective envelope of D/I, and . Recently Zhou, Kim, and Hu showed that is a commutative ring with identity containing D/I. The Krull domains are a w-analog of the Dedekind domains in the sense that D is a Krull domain if and only if every nonzero ideal of D is w-invertible. It is well known that D is a Dedekind domain if and only if D/aD is a zero-dimensional semilocal principal ideal ring (PIR) for all nonzero nonunits , if and only if D/I is a zero-dimensional semilocal PIR for all nonzero ideals I of D. In this paper, we prove that D is a Krull domain if and only if is a zero-dimensional semilocal PIR for any nonzero nonunit a of D, if and only if is a zero-dimensional semilocal PIR for any w-ideal I of D. We also study when is Noetherian, Artinian, or a (zero-dimensional) semilocal Bézout ring for all nonzero principal ideals or w-ideals I of D.