On locally divided integral domains and CPI-overrings

On locally divided integral domains and CPI-overrings

It is proved that an integral domain R is locally divided if and only if each CPI-extension of ℬ (in the sense of Boisen and Sheldon) is R-flat (equivalently, if and only if each CPI-extension of R is a localization of R). Thus, each CPI-extension of a locally divided domain is also locally divided....

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Bibliographic Details
Main Author: David E. Dobbs
Format: Article
Language:English
Published: Hindawi Limited 1981-01-01
Series:International Journal of Mathematics and Mathematical Sciences
Subjects:
integral domain
prime ideal
CPI-extension
flat over-ring
localization
locally divided
going-down
treed
quasilocal
QQR-property
Δ-domain
D+M construction
Krull direction.
Online Access:http://dx.doi.org/10.1155/S0161171281000082
Description
Summary:It is proved that an integral domain R is locally divided if and only if each CPI-extension of ℬ (in the sense of Boisen and Sheldon) is R-flat (equivalently, if and only if each CPI-extension of R is a localization of R). 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 2, and qusilocal domains with the QQR-property. The property of being CPI-closed behaves nicely with respect to the D+M construction, but is not a local property.
ISSN:0161-1712
1687-0425

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