Examples of Jaffard domains

Bouvier, Alain; Kabbaj, S.

doi:10.1016/0022-4049(88)90027-8

Cited by 20 publications

(18 citation statements)

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“…There is an old question (see [6]) asking if is it possible to find a UFD (or a Krull domain) which is not Jaffard. We note that if there exists a Krull domain which is not Jaffard, then we do have an example of a w-Jaffard domain which is not Jaffard.…”

Section: Examplesmentioning

confidence: 99%

“…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]). …”

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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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Section: Examplesmentioning

confidence: 99%

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confidence: 99%

W-Jaffard Domains in Pullbacks

Sahandi

2012

J. Algebra Appl.

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“…Hence from the above theorem, it can be seen that a Krull domain is w-Jaffard. There is an old question (see [5]) asking if is it possible to find a UFD (or a Krull domain) which is not Jaffard. So, the natural question is the following: is it possible to find a w-Jaffard non Jaffard domain?…”

Section: Semistar-valuative Dimensionmentioning

confidence: 99%

Semistar-Krull and valuative dimension of integral domains

Sahandi

2009

Ricerche mat.

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Abstract. Given a stable semistar operation of finite type ⋆ on an integral domain D, we show that it is possible to define in a canonical way a stable semistar operation of finite typeMoreover we define the semistar valuative dimension of the domain D, denoted by ⋆-dimv(D), to be the maximal rank of the ⋆-valuation overrings of D. We show that ⋆-dimv(D) = n if and only ifand equality holds if D is a ⋆-Noetherian domain or is a Prüfer ⋆-multiplication domain. We define the ⋆-Jaffard domains as domainsAs an application, ⋆-quasi-Prüfer domains are characterized as domains D such that each (⋆, ⋆ ′ )-linked overring T of D, is a ⋆ ′ -Jaffard domain, where ⋆ ′ is a stable semistar operation of finite type on T . As a consequence of this result we obtain that a Krull domain D, must be a w D -Jaffard domain.

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“…(C 1 ) turned out to be true in three large (presumably different) classes of commutative rings, namely, (a) Krulltype domains, e.g., unique factorization domains (UFDs) or Krull domains [22,17]; (b) pseudo-valuation domains of type n [17]; and (c) Jaffard domains [16,5]. A finite-dimensional domain R is said to be Jaffard if dim(R[X 1 , ..., X n ]) = n + dim(R) for all n ≥ 1; equivalently, if dim(R) = dim v (R) [1,4,14,19,27]. The class of Jaffard domains contains most of the well-known classes of finitedimensional rings involved in dimension theory of commutative rings, such as Noetherian domains [29], Prüfer domains [19], universally catenarian domains [3], and stably strong S-domains [28,30] [21,26,31,34]; as a matter of fact, these mainly arise as polynomial rings over Prüfer domains or as pullbacks, and both settings either yield Jaffard domains or turn out to be inconclusive (in terms of allowing the construction of counterexamples) [1,15].…”

Section: Introductionmentioning

confidence: 99%