Local–Global Properties for Semistar Operations#

Abstract

We study the “local” behavior of several relevant properties concerning semistar operations, like finite type, stable, spectral, e.a.b. and a.b. We deal with the “global” problem of building a new semistar operation on a given integral domain, by “gluing” a given homogeneous family of semistar operations defined on a set of localizations. We apply these results for studying the local–global behavior of the semistar Nagata ring and the semistar Kronecker function ring. We prove that an integral domain D is a Prüfer ⋆-multiplication domain if and only if all its localizations D P are Prüfer ⋆ P -multiplication domains.