On the cardinality of star operations on a pseudo-valuation domain

摘要

Let R be a pseudo-valuation domain with residue field k, and let V be the associated valuation domain of R with residue field L. The purpose of this article is to compute the cardinality |Star (R)| (respectively |SStar (R)|) of star (respectively semistar) operations on R. It depends upon the relation between the residue fields of R and V. We will show that |Star (R)| < ∞ if and only if dimk L = 1, 2, 3, or L is a finite field, and that |SStar (R)| < ∞ if and only if |Star (R)| < ∞ and dimR < ∞.