Directed Unions of Local Quadratic Transforms of Regular Local Rings and Pullbacks

Guerrieri, Lorenzo; Heinzer, William; Olberding, Bruce; Toeniskoetter, Matthew

doi:10.1007/978-3-319-65874-2_13

Cited by 6 publications

(14 citation statements)

References 31 publications

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“…We discuss some properties of this ring. At the end of this article, in Theorem 6.3 we are going to prove that this ring is a GCD domain showing that monoidal Shannon extensions can be GCD domains without being valuation domains, while this is impossible for quadratic Shannon extensions by [11,Theorem 6.2]. , and…”

Section: Definition 32 Given An Extension Of Integral Domainsmentioning

confidence: 99%

“…But instead it turns out to be true if we assume S ∈ M d−1 (R) and m S to be principal. To prove this we are going to use the characterization of quadratic Shannon extensions as pullbacks proved in [11].…”

Section: Monoidal Shannon Extensions With Principal Maximal Idealmentioning

confidence: 99%

“…Recently, authors like Heinzer, Loper, Olberding, Schoutens, Toeniskoetter ( [12], [13], [11]) and others studied the ideal-theoretic structure of quadratic Shannon extensions without focusing particularly on the geometric origin of these concepts. The tools used in these works are usually those from multiplicative ideal theory.…”

Section: Introductionmentioning

confidence: 99%

“…In [11,Theorem 6.2] is proved that a quadratic Shannon extension of a regular local ring is a GCD domain if and only if it is a valuation ring. In the monoidal case we find Shannon extensions which are GCD domains but not valuation domains.…”

Section: Introductionmentioning

confidence: 99%

“…The second class is formed by the domains having a finite intersection of principal ideals finitely generated only when equal to one of the principal ideals intersected (see Definitions 2.1 and 2.4). We characterize when a directed union of Noetherian GCD domains is in this second class, deriving as a consequence a generalization of [11,Theorem 6.2].…”

Section: Introductionmentioning

confidence: 99%

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Directed unions of local monoidal transforms and GCD domains

Guerrieri

2019

J. Algebra Appl.

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Let (R, m) be a regular local ring of dimension d ≥ 2. A local monoidal transform of R is a ring of the formIn this article we study some features of the rings S = ∪ ∞ n≥0 R n obtained as infinite directed union of iterated local monoidal transforms of R. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains. MSC: 13A15, 13F15, 13G05, 13H05.

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Section: Definition 32 Given An Extension Of Integral Domainsmentioning

confidence: 99%

Section: Monoidal Shannon Extensions With Principal Maximal Idealmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Directed unions of local monoidal transforms and GCD domains

Guerrieri

2019

J. Algebra Appl.

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The Quadratic Tree of a Two-Dimensional Regular Local Ring

Heinzer

Loper

Olberding

et al. 2023

Algebraic, Number Theoretic, and Topological Aspects of Ring Theory

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t-Local Domains and Valuation Domains

Fontana¹,

Zafrullah

2019

Trends in Mathematics

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In a valuation domain (V, M ) every nonzero finitely generated ideal J is principal and so, in particular, J = J t , hence the maximal ideal M is a t-ideal. Therefore, the t-local domains (i.e., the local domains, with maximal ideal being a t-ideal) are "cousins" of valuation domains, but, as we will see in detail, not so close. Indeed, for instance, a localization of a t-local domain is not necessarily t-local, but of course a localization of a valuation domain is a valuation domain.So it is natural to ask under what conditions is a t-local domain a valuation domain? The main purpose of the present paper is to address this question, surveying in part previous work by various authors containing useful properties for applying them to our goal. Dedicated to David F. Andersonshorter, more efficient proof of Theorem 6.2 of [27]. We also point to examples of maximal t-ideals Q in a particular domain D such that D Q is not t-local. Background results and t-local domainsWe start with proving some important preliminary results. But, for that, we need to recall the formal definition of star operation. A star operation on D is a map

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Directed Unions of Local Quadratic Transforms of Regular Local Rings and Pullbacks

Abstract: Let {R n , m n } n≥0 be an infinite sequence of regular local rings with R n+1 birationally dominating R n and m n R n+1 a principal ideal of R n+1 for each n. We examine properties of the integrally closed local domain S = n≥0 R n .

Cited by 6 publications

References 31 publications

Directed unions of local monoidal transforms and GCD domains

Directed unions of local monoidal transforms and GCD domains

The Quadratic Tree of a Two-Dimensional Regular Local Ring

t-Local Domains and Valuation Domains

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