Boundary valuation domains

Maney, Jack

doi:10.1016/s0021-8693(03)00433-2

Cited by 6 publications

(9 citation statements)

References 5 publications

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“…Following [16], let R be an HF D with quotient field K. If R = K, we define the boundary map δ R : K * → Z by δ R (α) = t − s, where α = (x 1 ...x t )/(y 1 ...y s ) ∈ K and x i , y j are irreducible elements in R.…”

Section: Atomic Pvdsmentioning

confidence: 99%

“…According to [4], an HF D is strongly half-factorial domain (SHF D) if each of its overrings is an HF D. An HF D is locally half-factorial domain (LHF D) if each of its localization is an HF D. Following [16], let R be an HF D with quotient field K. If R = K, we define the boundary map δ R : K * → Z by δ R (α) = t − s, where α = (x 1 ...x t )/(y 1 ...y s ) ∈ K and x i , y j are irreducible elements in R.…”

Section: Introductionmentioning

confidence: 99%

“…By [16], an integral domain R with quotient field K, is called boundary valuation domain (BV D) if R is an HF D, and for any α ∈ K with δ R (α) = 0, either α ∈ R or α −1 ∈ R, where δ R is boundary map defined on K.…”

Section: Introductionmentioning

confidence: 99%

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Note on Pseudo-Valuation Domains Which Are Not Valuation Domains

Shah¹,

Khan²

2014

International Electronic Journal of Algebra

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Abstract. In this article, we discuss the n-root closedness, root closedness, seminormality, S-root closedness, S-closedness, F -closedess of PVDs. A valuation domain, being integrally closed, is obviously root closed. So our interest of study is for a class of non-valuation PVDs. Let R ⊂ B be a domain extension such that R is a PVD and the common ideal P of R and B is a prime ideal in R. If R is n-root closed (respectively root closed, seminormal, S-root closed, S-closed, F -closed) in B, then R/P is PVD, which is n-root closed (respectively root closed, seminormal, S-root closed, S-closed, F -closed) in B/P .Further we study the relationship of atomic PVDs to atomic PVDs, SHFDs, LHFDs and BVDs. We also discuss a relative ascent and descent in general and particularly for the antimatter property of PVDs.Mathematics Subject Classification 2010: 13A15, 13A18, 13B30, 13F30

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Section: Atomic Pvdsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Note on Pseudo-Valuation Domains Which Are Not Valuation Domains

Shah¹,

Khan²

2014

International Electronic Journal of Algebra

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“…In [34], many interesting properties of the boundary valuation domains are explored, and insights into the overring behavior of HFDs are obtained. Additionally, boundary valuation domains are completely characterized using groups of divisibility.…”

Section: Conjecture 92mentioning

confidence: 99%

Extensions of Half-Factorial Domains

Coykendall¹

2005

Lecture Notes in Pure and Applied Mathematics

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A half-factorial domain (HFD) is an atomic domain, R, with the property that if one has the irreducible factorizations in RHalf-factorial domains were implicitly studied in the case of rings of algebraic integers in a 1960 paper of L. Carlitz [11] and were subsequently abstracted and studied by A. Zaks.Since their inception as a generalization of the classical notion of unique factorization domains, half-factorial domains have been the subject of much interest in commutative algebra. This paper will give a survey of some recent advances in the study of half-factorial domains with the emphasis on advances in the understanding of ring extension behavior of half-factorial domains.

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“…Given a domain R in which every nonzero element can be factored into unique lengths of irreducibles (in other words, R is a half factorial domain or HFD), R is a boundary valuation domain or BVD if every element of the quotient field of R with more irreducibles on the numerator is in R itself. That is to say, for every a b ∈ QF (R) with irreducible factorization π 1 ...πn η 1 ...ηm , with n > m, a b ∈ R. The interplay between PVDs and their valuation overrings was examined implicitly by Maney in [4]. In that paper, the class of all BVDs was characterized solely in terms of necessary and sufficient divisibility properties.…”

Section: Introductionmentioning

confidence: 99%

Abelian qo-groups and atomic pseudo-valuation domains

Stines

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Pseudo-valuation domains have been studied since their introduction in 1978 by Hedstrom and Houston. Related objects, boundary valuation domains, were introduced by Maney in 2004. Here, it is shown that the class of atomic pseudo-valuation domains coincides with the class of boundary valuation domains. It is also shown that power series rings and generalized power series rings give examples of pseudo-valuation domains whose congruence lattices can be characterized. The paper also introduces, and makes use of, a sufficient condition on the group of divisibility of a domain to guarantee that it is a pseudo-valuation domain.

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Boundary valuation domains

Cited by 6 publications

References 5 publications

Note on Pseudo-Valuation Domains Which Are Not Valuation Domains

Note on Pseudo-Valuation Domains Which Are Not Valuation Domains

Extensions of Half-Factorial Domains

Abelian qo-groups and atomic pseudo-valuation domains

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