Star Operations on Overrings and Semistar Operations#

Abstract

ABSTRACT The purpose of this article is to deepen the study of the relation between semistar operations on an integral domain D and the (semi) star operations (that is, the semistar operations, that restricted to the set of the fractional ideals, are star operations (on the overrings of D . First, we define the composition of two semistar operations and study when this composition is a semistar operation. Then we show that there is a bijection between the set of all semistar operations on a domain D and the set of all (semi) star operations on the overrings of D . To do this, we prove that semistar operations on D have a canonical decomposition as the composition of a semistar operation given by the extension to an overring and a (semi) star operation on this overring. Moreover, we study which properties of semistar operations are preserved by this bijection. Finally, we give some applications to the study of semistar operations on valuation and Prüfer domains and we give, by using the techniques introduced in this article, a characterization of generalized Dedekind domains in terms of the H-domains introduced by Glaz and Vasconcelos (1977).