Locally pseudo-valuation domains with only finitely many star operations

Let R be a locally pseudo-valuation domain. For each maximal ideal M of R  , denote by V(M)V(M) the associated valuation domain of the pseudo-valuation domain RMRM, and let T=⋂V(M)T=⋂V(M). We characterize those R   that have only finitely many star operations. When (R:T)≠(0)(R:T)≠(0), the characterization becomes: R has only finitely many star operations if and only if T   and each RMRM have only finitely many star operations. On the other hand, if (R:T)=(0)(R:T)=(0), then neither direction of this equivalence holds, and we give examples to illustrate the various possibilities.