Locally pseudo-valuation domains with only finitely many star operations

Abstract

Let R be a locally pseudo-valuation domain. For each maximal ideal M of R, denote by V(M) the associated valuation domain of the pseudo-valuation domain R-M, and let T = boolean AND V(M)" We characterize those R that have only finitely many star operations. When (R : T) not equal(0), the characterization becomes: R has only finitely many star operations if and only if T and each R-M have only finitely many star operations. On the other hand, if (R : T) = (0), then neither direction of this equivalence holds, and we give examples to illustrate the various possibilities. (C) 2015 Elsevier Inc. All reserved.