On integral domains whose overrings are Kaplansky ideal transforms

摘要

Let R be an integral domain with quotient field K. The Kaplansky transform of an ideal I of R is given by Omega (1) = {z epsilon K rad((R :(R) zR)) superset of or equal to I}. For finitely generated ideals, this agrees with the Nagata transform. We attempt to characterize Omega -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 Prufer domains: a Prufer domain R is an Omega -domain if and only if for each nonzero branched prime ideal P of R the set P-down arrow = {Q epsilon Spec(R)Q subset of or equal to P} is open in the Zariski topology. (C) 2001 Elsevier Science B.V. All rights reserved. [References: 15]