INTEGRALLY CLOSED DOMAINS WITH ONLY FINITELY MANY STAR OPERATIONS

Abstract

We prove that an integrally closed domain R admits only finitely many star operations if and only if R satisfies each of the following conditions: (1) R is a Prufer domain with finite character, (2) all but finitely many maximal ideals of R are divisorial, (3) only finitely many maximal ideals of R contain a nonzero prime ideal that is contained in some other maximal ideal of R, and (4) if P not equal (0) is the largest prime ideal contained in a (necessarily finite) collection of maximal ideals of R, then the prime spectrum of R/P is finite.