Sharpness and semistar operations in Prüfer-like domains

Abstract

Let * be a semistar operation on a domain D, *f the finite-type semistar operation associated to *, and D a Prüfer *-multiplication domain (P * MD). For the special case of a Prüfer domain (where * is equal to the identity semistar operation), we show that a nonzero prime P of D is sharp, that is, that (Formula presented.), where the intersection is taken over the maximal ideals M of D that do not contain P, if and only if two closely related spectral semistar operations on D differ. We then give an appropriate definition of *f-sharpness for an arbitrary P*MD D and show that a nonzero prime P of D is *f-sharp if and only if its extension to the *-Nagata ring of D is sharp. Calling a P*MD *f-sharp (*f-doublesharp) if each maximal (prime) *f-ideal of D is sharp, we also prove that such a D is *f-doublesharp if and only if each (*, t) -linked overring of D is *f-sharp.