Let D be an integral domain and ? a semistar operation on D. As a generalization of the notion of Noetherian domains to the semistar setting, we say that D is a ?-Noetherian domain if it has the ascending chain condition on the set of its quasi-?-ideals. On the other hand, as an extension the notion of Pr ufer domain (and of Pr ufer v-multiplication domain), we say that D is a Pr ufer ?-multiplication domain (P?MD, for short) if DM is a valuation domain, for each quasi-? f -maximal ideal M of D. Finally, recalling that a Dedekind domain is a Noetherian Pr ufer domain, we deÿne a ?-Dedekind domain to be an integral domain which is ?-Noetherian and a P?MD. For the identity semistar operation d, this deÿnition coincides with that of the usual Dedekind domains and when the semistar operation is the v-operation, this notion gives rise to Krull domains. Moreover, Mori domains not strongly Mori are ?-Dedekind for a suitable spectral semistar operation.Examples show that ?-Dedekind domains are not necessarily integrally closed nor onedimensional, although they mimic various aspects, varying according to the choice of ?, of the "classical" Dedekind domains. In any case, a ?-Dedekind domain is an integral domain D having a Krull overring T (canonically associated to D and ?) such that the semistar operation ? is essentially "univocally associated" to the v-operation on T .In the present paper, after a preliminary study of ?-Noetherian domains, we investigate the ?-Dedekind domains. We extend to the ?-Dedekind domains the main classical results and several characterizations proven for Dedekind domains. In particular, we obtain a characterization of a ?-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 ?-Dedekind domain if and only if the Nagata semistar domain Na(D; ?) is a Dedekind domain. Several applications of the general results are given for special cases of the semistar operation ?.
We introduce and study the notion of -stability with respect to a semistar operation defined on a domain R; in particular we consider the case where is the w-operation. This notion allows us to generalize and improve several properties of stable domains and totally divisorial domains. MSC: Primary: 13A15; secondary: 13F05; 13G05 IntroductionStar operations, such as the v-closure (or divisorial closure), the t-closure and the w-closure, are an essential tool in modern multiplicative ideal theory for characterizing and investigating several classes of integral domains. For example, in the last few decades a large amount of literature has appeared on Mori domains, that is domains satisfying the ascending chain condition on divisorial ideals, and Prüfer v-multiplication domains, for short PvMDs, that is domains in which each finitely generated ideal is t-invertible (or w-invertible). The consideration that some important operations on ideals, like the integral closure, satisfy almost all the properties of star operations led Okabe and Matsuda to introduce in 1994 the more general and flexible notion of semistar operation [26]. The class of semistar operations includes the classical star operations and often provides a more appropriate context for approaching several questions of multiplicative ideal theory; see for example [10,[14][15][16][17]32]. In this paper, we introduce the notion of -stability with respect to a semistar operation . Motivated by earlier work of Bass [4] and Lipman [25] on the number of generators of an ideal, in 1974Sally and Vasconcelos defined a Noetherian ring R to be stable if each nonzero ideal of R is projective over its endomorphism ring End R (I ) [35]. In a note of 1987, Anderson, Huckaba and Papick considered the notion of stability for arbitrary integral domains [2]. When I is a nonzero ideal of a domain R, then End R (I ) = (I : I ); thus a domain R is stable if each nonzero ideal I of R is invertible in the overring (I : I ). Since 1998, stable domains have been thoroughly investigated by Olberding in a series of papers [27][28][29][30][31].Given a semistar operation on a domain R, we say that a nonzero ideal I of R is -stable if I is˙ -invertible in (I : I ) and that R is -stable if each nonzero ideal of R is -stable. (Here we denote by˙ the semistar operation
Abstract. 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 ⋆-invertibility and quasi-⋆-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. Among the main properties proved here, we mention the following: (a) several characterizations of ⋆-invertibility and quasi-⋆-invertibility and necessary and sufficient conditions for the equivalence of these two notions; (b) the relations between the ⋆-invertibility (or quasi-⋆-invertibility) and the invertibility (or quasi-invertibility) with respect to the semistar operation of finite type (denoted by ⋆ f ) and to the stable semistar operation of finite type (denoted by ⋆), canonically associated to ⋆; (c) a characterization of the H(⋆)-domains in terms of semistar-invertibility (note that the H(⋆)-domains generalize, in the semistar setting, the H-domains introduced by Glaz and Vasconcelos); (d) for a semistar operation of finite type a nonzero finitely generated (fractional) ideal I is ⋆-invertible (or, equivalently, quasi-⋆-invertible, in the stable semistar case) if and only if its extension to the Nagata semistar ring I Na(D, ⋆) is an invertible ideal of Na(D, ⋆).
Abstract. 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.
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