Star stable domains

Abstract

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

w-Stability and Cliffordw-Regularity of Polynomial Rings

“…When Stability with respect to star operations (and more generally to semistar operations) was introduced and studied by the authors of this paper in [16].…”

“…Recall that a Prüfer domain (resp., a PvMD) is called strongly discrete if P = P 2 for each prime (resp., t-prime) ideal P . A domain is integrally closed and Clifford w-regular (resp., w-stable) if and only if it is a PvMD (resp., a strongly discrete PvMD) with t-finite character [17,Corollary 4.5], [16,Therem 2.9]. Thus, for w = d, an integrally closed Clifford regular (resp., stable) domain is precisely a Prüfer domain (resp., a strongly discrete Prüfer domain) with finite character [5,Theorem 4.5], [32,Theorem 4.6].…”

Star Stability and Star Regularity for Mori Domains

“…Here we mainly consider the cases where * = d, w, t. In fact the most interesting results on star stability and star regularity were obtained in [15] and [16] for star operations spectral and of finite type. In addition, if * is spectral and of finite type, * -regularity implies * = w [16,Corollary 1.7]; in particular, if R is Clifford regular, then w = d.…”

Finite character, local stability property and local invertibility property

“…In my conference talk, based on [8,9,10], I showed that many properties of stability and Clifford regularity can be generalized and improved in the set of (semi)star operations. In particular, by using the techniques developed in [9], the results illustrated in the next section can be proved for star regularity with respect to any star operation that is spectral and of finite type (see [9] for the relevant definitions).…”