On Local ☆-Completely Integrally Closed Domains

Abstract

Let ☆ be a star operation on an integral domain R. The domain R is a ☆-CICD if (AA ⁻¹)^☆ = R for all nonzero (fractional) ideals A of R. In this article, we prove that, if the maximal ideal of a local ☆-CICD is a ☆-ideal, then R is ☆-principal ideal domain. We also establish that any ☆-CICD R is locally a PID when ☆ is induced by the localizations at prime ideals of R.

Key Words:

• Completely integrally closed domains
• Krull domains
• Star operations

2010 Mathematics Subject Classification:

• Primary 13A15
• 13A18
• 16W50

ACKNOWLEDGMENTS

The author wishes to express his gratitude to Marco Fontana for pointing out some articles in the references of this article.

Notes

Communicated by U. Walther.