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On locally divided integral domains and CPI-overrings

DOI: 10.1155/s0161171281000082

Keywords: 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.

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Abstract:

It is proved that an integral domain R is locally divided if and only if each CPI-extension of ¢ (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.

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