International Journal of Mathematics and Mathematical Sciences
Volume 4 (1981), Issue 1, Pages 119-135
On locally divided integral domains and CPI-overrings
University of Tennessee, Knoxville 37916, Tennessee, USA
Received 7 March 1979
Copyright © 1981 David E. Dobbs. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
It is proved that an integral domain is locally divided if and only if each CPI-extension of (in the sense of Boisen and Sheldon) is -flat (equivalently, if and only if each CPI-extension of is a localization of ). Thus, each CPI-extension of a locally divided domain is also locally divided. Treed domains are characterized by the going-down behavior of their CPI-extensions. A new class of (not necessarily treed) domains, called CPI-closed domains, is introduced. Examples include locally divided domains, quasilocal domains of Krull dimension , and qusilocal domains with the QQR-property. The property of being CPI-closed behaves nicely with respect to the construction, but is not a local property.