Essential domains and two conjectures in dimension theory

Fontana, Marco; Kabbaj, S.

doi:10.1090/s0002-9939-04-07502-1

Cited by 18 publications

(5 citation statements)

References 27 publications

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“…It is an open problem to compute the dimension of polynomial rings over Krull domains in general. In this vein, Bouvier conjectured that "finitedimensional Krull (or more particularly factorial) domains need not be Jaffard" [8,15]. In Figure 1, a diagram of implications places this conjecture in its proper perspective and hence shows how it naturally arises.…”

Section: Diagram Of Implicationsmentioning

confidence: 97%

Bouvier's conjecture

Bouchiba,

Kabbaj

2008

Preprint

Self Cite

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Section: Diagram Of Implicationsmentioning

confidence: 97%

Bouvier's conjecture

Bouchiba,

Kabbaj

2008

Preprint

Self Cite

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“…It is an open problem to compute the dimension of polynomial rings over Krull domains in general. In this vein, Bouvier conjectured that "…nite-dimensional Krull (or, more particularly, factorial) domains need not be Ja¤ard" [18,44]. Bouvier's conjecture makes sense beyond the Noetherian context.…”

Section: Problem 37mentioning

confidence: 99%

Open Problems in Commutative Ring Theory

Cahen¹,

Fontana

Frisch

et al. 2014

Commutative Algebra

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This article consists of a collection of open problems in commutative algebra. The collection covers a wide range of topics from both Noetherian and non-Noetherian ring theory and exhibits a variety of research approaches, including the use of homological algebra, ring theoretic methods, and star and semistar operation techniques. The problems were contributed by the authors and editors of this volume, as well as other researchers in the area.

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“…In the current paper, we offer direct and self-contained proofs using techniques and basic results from commutative ring theory. Early and recent developments on prime spectra and dimension theory are to be found in [3,5,6,7,29,30,31,34,35] for the special case of tensor products of k-algebras, and in [1,11,17,22,24,26,27] for the general case. Any unreferenced material is standard, as in [24,26].…”

Section: Introductionmentioning

confidence: 99%

Embedding Dimension and Codimension of Tensor Products of Algebras over a Field

Bouchiba

Kabbaj

2017

Rings, Polynomials, and Modules

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Abstract. Let k be a field. This paper investigates the embedding dimension and codimension of Noetherian local rings arising as localizations of tensor products of k-algebras. We use results and techniques from prime spectra and dimension theory to establish an analogue of the "special chain theorem" for the embedding dimension of tensor products, with effective consequence on the transfer or defect of regularity as exhibited by the (embedding) codimension given by codim(R) := embdim(R) − dim(R).

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Essential domains and two conjectures in dimension theory

Abstract: Abstract. This note investigates two long-standing conjectures on the Krull dimension of integer-valued polynomial rings and of polynomial rings in the context of (locally) essential domains.

Cited by 18 publications

References 27 publications

Bouvier's conjecture

Bouvier's conjecture

Open Problems in Commutative Ring Theory

Embedding Dimension and Codimension of Tensor Products of Algebras over a Field

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