Universally Catenarian Integral Domains, Strong S-Domains and Semistar Operations

Abstract

Abstract. Let D be an integral domain and ⋆ a semistar operation stable and of finite type on it. In this paper, we are concerned with the study of the semistar (Krull) dimension theory of polynomial rings over D. We introduce and investigate the notions of ⋆-universally catenarian and ⋆-stably strong S-domains and prove that, every ⋆-locally finite dimensional Prüfer ⋆-multiplication domain is ⋆-universally catenarian, and this implies ⋆-stably strong S-domain. We also give new characterizations of ⋆-quasi-Prüfer domains introduced recently by Chang and Fontana, in terms of these notions. IntroductionThe concepts of S(eidenberg)-domains and strong S-domains are crucial ones and were introduced by Kaplansky [18, Page 26]. Recall that an integral domain D is an S-domain if for each prime ideal P of D of height one the extension P D[X] to the polynomial ring in one variable is also of height one. A strong S-domain is a domain D such that, D/P is an S-domain, for each prime P of D. One of the reasons why Kaplansky introduced the notion of strong S-domain was to treat the classes of Noetherian domains and Prüfer domains in a unified frame. Moreover, if D belongs to one of the two classes of domains, then the following dimension formula holds: dim(D[X 1 , · · · , X n ]) = n + dim(D) (cf., [23, Theorem 9] and [24, Theorem 4] is a strong S-domain for each n ≥ 1. Note that the class of Jaffard domains contains the class of stably strong S-domains. The class of stably strong S-domains contains an important class of universally catenarian domains. Recall that a domain D, is called catenarian, if for each pair P ⊂ Q of prime ideals of D, any two saturated chain of prime ideals between P and Q have the same finite length. If for each n ≥ 1, the polynomial ring D[X 1 , · · · , X n ] is catenary, then D is said to be universally catenarian (cf. [4,3]).For several decades, star operations, as described in [17, Section 32], have proven to be an essential tool in multiplicative ideal theory, for studying various classes of 2000 Mathematics Subject Classification. Primary 13G05, 13B24, 13A15, 13C15. Key words and phrases. Semistar operation, star operation, Krull dimension, strong S-domain, Jaffard domain, universally catenarian, catenary. This manuscript is a sequel to [22]. Given a semistar operation ⋆ on D and let ⋆ be the stable semistar operation of finite type canonically associated to ⋆ (the definitions are recalled later in this section), it is possible to define a semistar operation stable and of finite typefor each positive integer n. Every ⋆-Noetherian and P⋆MDs are ⋆-Jaffard domains (cf. [22]). In this paper we define and study two subclass of ⋆-Jaffard domains. Namely in Sections 2 and 3, we define and study ⋆-stably strong S-domains and ⋆-universally catenarian domains. In Section 4 we give new characterizations of ⋆-quasi-Prüfer domains in terms of ⋆-stably strong S-domains and ⋆-universally catenarian domains.To facilitate the reading of the introduction and of the paper, we first review some basic facts on semi...

On Quasi-Prüfer and UMtDomains

“…where the first equality is by [16,Theorem 4.5]. Finally by [17,Corollary 2.6], every UMt domain of finite w-dimension is a w-Jaffard domain to deduce that T is a w-Jaffard domain.…”

“…In [16] we defined and studied the w-Jaffard domains and proved that all strong Mori domains (domains that satisfy the ACC on w-ideals) and all UMt domains of finite w-dimension, are w-Jaffard domains. In [17] we defined and studied a subclass of w-Jaffard domains, namely the w-stably strong S-domains and showed how this notion permit studies of UMt domains in the spirit of earlier works on quasi-Prüfer domains. The aim of this paper is to prove that, for a domain D with some condition on w-dim(D), the following statements are equivalent, which gives new descriptions of quasi-Prüfer domains; a result reminiscent of the well-known result of Ayache, Cahen and Echi [2] (see also [9, Theorem 6.7.8]).…”

On S-Semistar-Noetherian Domains

“…(iii) For every ideal I of D, Sat S (I) w = (I w : D t) for some t ∈ S. Now we prove a Hilbert Basis Theorem for S-˜ -Noetherian domains. To this purpose we use the semistar operation [X] on D[X], induced canonically from the semistar operation on D, introduced by the second author [17] (see also [18]).…”

W-Jaffard Domains in Pullbacks

“…Note that, the notions of * -dimension, t-dimension, and of w-dimension have received a considerable interest by several authors (cf. for instance, [22,23,24,14,15,28,29]). Now we recall a special case of a general construction for semistar operations (see [22]).…”