Stability and Clifford Regularity with Respect to Star Operations

Abstract

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.

w-Stability and Cliffordw-Regularity of Polynomial Rings

“…Then Halter-Koch, in the language of ideal systems, introduced Clifford * -regularity for star operations of finite type [22]. Finally, we deepened the study of stability and Clifford regularity with respect to star operations in [17,18].…”

“…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].…”

“…(2) Let D be of t-dimension one. Since a Clifford w-regular domain has t-finite character [17,Theorem 5.2], D is weakly Matlis and so D[X] is weakly Matlis by item (1). Besides, each upper to zero of D[X] is t-maximal (of height one).…”

“…[17, Proposition 5.4 and Theorem 5.6] Let D be an integral domain. Assume that (a) D is integrally closed or (b) each nonzero (t-)prime ideal of D is contained in a unique (t-)maximal ideal (e.g., D has (t-)dimension one).…”

Star Stability and Star Regularity for Mori Domains

“…The first attempt to extend the notion of Clifford regularity in the setting of star operations is due to S. Kabbaj and A. Mimouni, who studied Clifford t-regularity for PvMDs and Mori domains [20,21,22,23]. Successively F. Halter-Koch, in the language of ideal systems, introduced Clifford * -regularity for any star operation of finite type [18] and the authors of this paper continued this study in [16].…”