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Let  D  be an integral domain,  w  be the so-called  w -operation on  D , andbe 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 thatis 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 ifis a zero-dimensional semilocal PIR for any nonzero nonunit  a  of  D , if and only ifis a zero-dimensional semilocal PIR for any  w -ideal  I  of  D . We also study whenis Noetherian, Artinian, or a (zero-dimensional) semilocal Bézout ring for all nonzero principal ideals or  w -ideals  I  of  D .

A characterization of Krull domains in terms of their factor rings