arXiv subjects
Publications and source records attributed to Giampaolo Picozza.
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.
In the last few years, the concepts of stability and Clifford regularity have been fruitfully extended by using star operations. In this paper we deepen the study of star stable and star regular domains and relate these two classes of domains to each other.
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.
Let $D$ be an integral domain with quotient field $K$. The $b$-operation that associates to each nonzero $D$-submodule $E$ of $K$, $E^b := \bigcap\{EV \mid V valuation overring of D\}$, is a semistar operation that plays an important role in many questions of ring theory (e.g., if $I$ is a nonzero ideal in $D$, $I^b$ coincides with its integral closure). In a first part of the paper, we study the integral domains that are $b$-Noetherian (i.e., such that, for each nonzero ideal $I$ of $D$, $I^b = J^b$ for some a finitely generated ideal $J$ of $D$). For instance, we prove that a $b$-Noetherian domain has Noetherian spectrum and, if it is integrally closed, is a Mori domain, but integrally closed Mori domains with Noetherian spectra are not necessarily $b$-Noetherian. We also characterize several distinguished classes of $b$-Noetherian domains. In a second part of the paper, we study more generally the e.a.b. semistar operation of finite type $\star_a$ canonically associated to a given semistar operation $\star$ (for instance, the $b$-operation is the e.a.b. semistar operation of finite type canonically associated to the identity operation). These operations, introduced and studied by Krull, Jaffard, Gilmer and Halter-Koch, play a very important role in the recent generalizations of the Kronecker function ring. In particular, in the present paper, we classify several classes of integral domains having some of the fundamental operations $d$, $t$, $w$ and $v$ equal to some of the canonically associated e.a.b. operations $b$, $t_a$, $w_a$ and $v_a$.
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.
We introduce the concept of \textit{quasi-stable} ideal in an integral domain $D$ (a nonzero fractional ideal $I$ of $D$ is quasi-stable if it is flat in its endomorphism ring $(I \colon I)$) and study properties of domains in which each nonzero fractional ideal is quasi-stable. We investigate some questions about flatness that were raised by S. Glaz and W.V. Vasconcelos in their 1977 paper \cite{GV}.
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.
The purpose of this paper is to deepen the study of the Prüfer $\star$--multiplication domains, where $\star$ is a semistar operation. For this reason, in Section 2, we introduce the $\star$--domains, as a natural extension of the $v$--domains, where $v$ is the classical Artin's divisorial operation. We investigate their close relation with the Prüfer $\star$--multiplication domains. In particular, in Section 3, we obtain a characterization of Prüfer $\star$--multiplication domains in terms of $\star$--domains satisfying a variety of coherent-like conditions. In Section 4, we extend to the semistar setting the notion of $\texttt{H}$--domain introduced by Glaz and Vasconcelos and we show, among the other results that, in the class of the $\texttt{H}(\star)$--domains, the Prüfer $\star$--multiplication domains coincide with the $\star$--domains.
We introduce and study the notion of $\star$-stability with respect to a semistar operation $\star$ defined on a domain $R$; in particular we consider the case where $\star$ is the $w$-operation. This notion allows us to generalize and improve several properties of stable domains and totally divisorial domains.
After the introduction in 1994, by Okabe and Matsuda, of the notion of semistar operation, many authors have investigated different aspects of this general and powerful concept. A natural development of the recent work in this area leads to investigate the concept of invertibility in the semistar setting. In this paper, we will show the existence of a ``theoretical obstruction'' for extending many results, proved for star-invertibility, to the semistar case. For this reason, we will introduce two distinct notions of invertibility in the semistar setting (called $\star$--invertibility and quasi--$\star$--invertibility), we will discuss the motivations of these ``two levels'' of invertibility and we will extend, accordingly, many classical results proved for the $d$--, $v$--, $t$-- and $w$-- invertibility.
Let $D$ be an integral domain and $\star$ a semistar operation on $D$. As a generalization of the notion of Noetherian domains to the semistar setting, we say that $D$ is a $\star$--Noetherian domain if it has the ascending chain condition on the set of its quasi--$\star$--ideals. On the other hand, as an extension the notion of Prüfer domain (and of Prüfer $v$--multiplication domain), we say that $D$ is a Prüfer $\star$--multiplication domain (P$\star$MD, for short) if $D_M$ is a valuation domain, for each quasi--$\star_{_{f}}$--maximal ideal $M$ of $D$. Finally, recalling that a Dedekind domain is a Noetherian Prüfer domain, we define a $\star$--Dedekind domain to be an integral domain which is $\star$--Noetherian and a P$\star$MD. In the present paper, after a preliminary study of $\star$--Noetherian domains, we investigate the $\star$--Dedekind domains. We extend to the $\star$--Dedekind domains the main classical results and several characterizations proven for Dedekind domains. In particular, we obtain a characterization of a $\star$--Dedekind domain by a property of decomposition of any semistar ideal into a ``semistar product'' of prime ideals. Moreover, we show that an integral domain $D$ is a $\star$--Dedekind domain if and only if the Nagata semistar domain Na$(D, \star)$ is a Dedekind domain. Several applications of the general results are given for special cases of the semistar operation $\star$.