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On Local ☆-Completely Integrally Closed Domains

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.