Semistar-Krull and valuative dimension of integral domains

Sahandi, Parviz

doi:10.1007/s11587-009-0059-8

Cited by 8 publications

(26 citation statements)

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“…It is shown in [22,Theorem 4.14] that, a UMt domain of finite w-dimension is a w-Jaffard domain. Now we give an example of a w-Jaffard non UMt domain.…”

Section: Pullbacksmentioning

confidence: 99%

“…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]).…”

Section: Introductionmentioning

confidence: 99%

“…for instance, [22,23,24,14,15,28,29]). Now we recall a special case of a general construction for semistar operations (see [22]). Let X, Y be two indeterminates over R, and let K := qf (R).…”

Section: Introductionmentioning

confidence: 99%

“…Ideal w-multiplication converts ring notions such as Dedekind, Noetherian, Prüfer, and quasi-Prüfer, respectively to Krull, strong Mori, PvMD, and UMt. As the wcounterpart of Jaffard domains, in [22], we introduced the class of w-Jaffard domains, as integral domains R such that w-dim(R) = w-dim v (R) < ∞. In this paper we study the transfer of w-Jaffard domains in pullback constructions, in order to provide original examples.…”

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confidence: 99%

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W-Jaffard Domains in Pullbacks

Sahandi

2012

J. Algebra Appl.

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Abstract. In this paper we study the class of w-Jaffard domains in pullback constructions, and give new examples of these domains. In particular we give examples to show that the two classes of w-Jaffard and Jaffard domains are incomparable. As another application, we establish that for each pair of positive integers (n, m) with n + 1 ≤ m ≤ 2n + 1, there is an (integrally closed) integral domain R such that w-dim(R) = n and w[X]-dim(R[X]) = m. IntroductionThroughout this paper, R denotes a (commutative integral) domain with identity with quotient field qf (R), and let X be an algebraically independent indeterminate over R. In [26, Theorem 2] Seidenberg proved that if R has finite Krull dimension, Krull [18] has shown that if R is any finite-dimensional Noetherian ring, then dim(R[X]) = 1 + dim(R) (cf. also [26, Theorem 9]). Seidenberg subsequently proved the same equality in case R is any finite-dimensional Prüfer domain. To unify and extend such results on Krull-dimension, Jaffard [17] introduced and studied the valuative dimension denoted by dim v (R), for a domain R. This is the maximum of the ranks of the valuation overrings of R. Jaffard proved in [17, Chapitre IV] that, if R has finite valuative dimension, then dim v (R[X]) = 1 + dim v (R), and that if R is a Noetherian or a Prüfer domain, then dim(R) = dim v (R). In [1] Anderson, Bouvier, Dobbs, Fontana and Kabbaj introduced the notion of Jaffard domains, as finite dimensional integral domains R such that dim(R) = dim v (R), and studied this class of domain systematically (see also [6]).The v, t and w-operations in integral domains are of special importance in multiplicative ideal theory and was investigated by many authors in the 1980's. Ideal w-multiplication converts ring notions such as Dedekind, Noetherian, Prüfer, and quasi-Prüfer, respectively to Krull, strong Mori, PvMD, and UMt. As the wcounterpart of Jaffard domains, in [22], we introduced the class of w-Jaffard domains, as integral domains R such that w-dim(R) = w-dim v (R) < ∞. In this paper we study the transfer of w-Jaffard domains in pullback constructions, in order to provide original examples.We need to recall some notions from star operations. Let F (R) denotes the set of nonzero fractional ideals, and f (R) be the set of all nonzero finitely generated fractional ideals of R. Let * be a star operation on the domain R. For every A ∈ F (R), put A * f := F * , where the union is taken over all F ∈ f (R) with2000 Mathematics Subject Classification. Primary 13G05, 13A15, 13C15.

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“…It is shown in [22,Theorem 4.14] that, a UMt domain of finite w-dimension is a w-Jaffard domain. Now we give an example of a w-Jaffard non UMt domain.…”

Section: Pullbacksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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W-Jaffard Domains in Pullbacks

Sahandi

2012

J. Algebra Appl.

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“…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 type ⋆[X] on D[X] (cf.…”

Section: Introductionmentioning

confidence: 99%

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

Sahandi

2010

Communications in Algebra

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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...

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An Overring-Theoretic Approach to Polynomial Extensions of Star and Semistar Operations

Chang

Fontana

2011

Communications in Algebra

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Abstract. Call a semistar operation ⊛ on the polynomial domain D[X] an extension (respectively, a strict extension) of a semistar operation ⋆ defined on an integral domain D, with quotient fieldIn this paper, we study the general properties of the above defined extensions and link our work with earlier efforts, centered on the stable semistar operation case, at defining semistar operations on D[X] that are "canonical" extensions (or, "canonical" strict extensions) of semistar operations on D.

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Semistar-Krull and valuative dimension of integral domains

Cited by 8 publications

References 26 publications

W-Jaffard Domains in Pullbacks

W-Jaffard Domains in Pullbacks

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

An Overring-Theoretic Approach to Polynomial Extensions of Star and Semistar Operations

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