When the Semistar Operation is the Identity

Picozza, Giampaolo; Tartarone, Francesca

doi:10.1080/00927870801941895

Cited by 20 publications

(14 citation statements)

References 36 publications

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“…We doubt that this is true but have no counterexample. (2) Picozza and Tartarone [28,Theorem 3.7] prove that a DW-domain that is both integrally closed and satisfies the finite-conductor property must be a Prüfer domain. (A domain E is a finite conductor domain if (a) ∩ (b) is finitely generated for all a, b ∈ E.) The proof involves two steps: an integrally closed finite conductor domain is a PVMD, and a PVMD that is also a DW-domain must be a Prüfer domain.…”

Section: Examplesmentioning

confidence: 99%

“…If fact, these are examples of DW-domains, that is, domains in which the two star operations d and w (reviewed below) coincide. DW-domains were introduced (but not named) in [7] and further studied in [8] (where they were called t-linkative domains), [25,27,28]. It is easy to see that D is a DW-domain if and only if I is principal for each finitely generated ideal I of D such that I v is principal (see [27,Proposition 2.1]).…”

Section: Introductionmentioning

confidence: 99%

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Integral domains in which any two v-coprime elements are comaximal

Houston

Zafrullah

2015

Journal of Algebra

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Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Integral domains in which any two v-coprime elements are comaximal

Houston

Zafrullah

2015

Journal of Algebra

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“…Recently, the class of DW-domains (that is, domains in which the d-operation and w-operations coincide) has received a good deal of attention in the papers [31,34,35]. In fact, it is known that these domains were already known many years before (Dobbs et al [8,9,10]) under the name of t-linkative domains.…”

Section: Dg-domainsmentioning

confidence: 99%

“…Later, the localization of DW-domains was also studied in [35]. We next consider the localization in connection with the DG-property.…”

Section: Corollary 23 (Cf [35 Corollary 33])mentioning

confidence: 99%

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A Half-Centered Star-Operation on an Integral Domain

Wang¹,

Qiao²

2017

Journal of the Korean Mathematical Society

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Abstract. In this paper, we study the natural star-operation defined by the set of associated primes of principal ideals of an integral domain, which is called the g-operation. We are mainly concerned with the idealtheoretic properties of this star-operation. In particular, we investigate DG-domains (i.e., integral domains in which each ideal is a g-ideal), which form a proper subclass of the DW-domains. In order to provide some original examples, we examine the transfer of the DG-property to pullbacks. As an application of the g-operation, it is shown that w-divisorial Mori domains can be seen as a Gorenstein analogue of Krull domains.

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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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When the Semistar Operation is the Identity

Cited by 20 publications

References 36 publications

Integral domains in which any two v-coprime elements are comaximal

Integral domains in which any two v-coprime elements are comaximal

A Half-Centered Star-Operation on an Integral Domain

t-Local Domains and Valuation Domains

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