Abstract
Let be a semistar operation on a domain D,
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
, 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
-sharpness for an arbitrary P
MD D and show that a nonzero prime P of D is
-sharp if and only if its extension to the
-Nagata ring of D is sharp. Calling a P
MD
-sharp (
-doublesharp) if each maximal (prime)
-ideal of D is sharp, we also prove that such a D is
-doublesharp if and only if each
-linked overring of D is
-sharp.
2010 MATHEMATICS SUBJECT CLASSIFICATION:
Acknowledgements
The authors thank the referee for his/her comments and suggestions, which contributed to improving the final version of this article.
Disclosure statement
No potential conflict of interest was reported by the authors.