Giampaolo Picozza

Semistar Invertibility on Integral Domains

A natural development of the recent work about semistar operations leads to investigate the concept of “semistar invertibility”. Here, we show the existence of a “theoretical obstruction” for extending many results, proved for star-invertibility, to the semistar case. For this reason, we introduce two distinct notions of invertibility in the semistar setting (called ⋆-invertibility and quasi-⋆-invertibility), we discuss the motivations of these “two levels” of invertibility and we extend, accordingly, many classical results proved for the d-, v-, t- and w-invertibility.

Star Operations on Overrings and Semistar Operations

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).

When the Semistar Operation is the Identity

We study properties of integral domains in which it is given a semistar operation ★ such that is the identity. In particular, we put attention to the case ★ = v, where v is the divisorial closure.

w-Divisoriality in Polynomial Rings

We extend the Bass–Matlis characterization of local Noetherian divisorial domains to the non-Noetherian case. This result is then used to study the following question: If a domain D is w-divisorial, that is, if each w-ideal of D is divisorial, then is D[X] automatically w-divisorial? We show that the answer is yes if D is either integrally closed or Mori.

STAR-INVERTIBILITY AND t-FINITE CHARACTER IN INTEGRAL DOMAINS

Let A be an integral domain. We study new conditions on families of integral ideals of A in order to get that A is of t-finite character (i.e. each nonzero element of A is contained in finitely many t-maximal ideals). We also investigate problems connected with the local invertibility of ideals.

Star Stability and Star Regularity for Mori Domains

In the last few years, the concepts of stability and Clifford regularity have been fruitfully extended by using star operations. In this paper we study and put in relation these properties for Noetherian and Mori domains, substantially improving several results present in the literature.

Stability and Clifford-regularity with respect to star operations

In the last few years, the concepts of stability and Clifford regularity have been fruitfully extended by using star operations. In this article we deepen the study of star stable and star regular domains and relate these two classes of domains to each other.

w-Stability and Clifford w-Regularity of Polynomial Rings

We investigate the transfer of w-stability and Clifford w-regularity from a domain D to the polynomial ring D[X]. We show that these two properties pass from D to D[X] when D is either integrally closed or it is Mori and w-divisorial.