On integral domains whose overrings are Kaplansky ideal transforms

Let R be an integral domain with quotient field K. The Kaplansky transform of an ideal I of R is given by Ω(I)={z set membership, variant Krad((R:RzR))superset of or equal toI}. For finitely generated ideals, this agrees with the Nagata transform. We attempt to characterize Ω-domains, that is, domains each of whose overrings is a Kaplansky transform. We obtain a particularly satisfactory characterization when we restrict to the class of Prüfer domains: a Prüfer domain R is an Ω-domain if and only if for each nonzero branched prime ideal P of R the set P↓={Qset membership, variantSpec(R)Qsubset of or equal toP} is open in the Zariski topology.