
On locally divided integral domains and CPIoverringsDOI: 10.1155/s0161171281000082 Keywords: integral domain , prime ideal , CPIextension , flat overring , localization , locally divided , goingdown , treed , quasilocal , QQRproperty , Δdomain , D+M construction , Krull direction. Abstract: It is proved that an integral domain R is locally divided if and only if each CPIextension of ￠ (in the sense of Boisen and Sheldon) is Rflat (equivalently, if and only if each CPIextension of R is a localization of R). Thus, each CPIextension of a locally divided domain is also locally divided. Treed domains are characterized by the goingdown behavior of their CPIextensions. A new class of (not necessarily treed) domains, called CPIclosed domains, is introduced. Examples include locally divided domains, quasilocal domains of Krull dimension 2, and qusilocal domains with the QQRproperty. The property of being CPIclosed behaves nicely with respect to the D+M construction, but is not a local property.
