Rings, Modules, and Closure Operations

Elliott, Jesse

doi:10.1007/978-3-030-24401-9

Cited by 27 publications

(13 citation statements)

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“…Thus Corollary 2 gives for ⋆ = t a t-Dedekind domain and Corollary 3 gives the name of a v-Dedekind domain to the G-Dedekind domain of [24] and Pseudo Dedekind domain of [5]. But there is a slight problem with this naming system, Jesse Elliott in [11] calls a Krull domain a t-Dedekind domain. So, perhaps, ⋆-G-Dedekind may be the general name with the note that a d-G-Dedekind domain is the usual Dedekind domain and a t-G-Dedekind domain is a locally factorial Krull domain while the v-G-Dedekind domain is the usual Pseudo Dedekind domain, or the old G-Dedekind domain.…”

Section: Of Course By Saying That An Idealmentioning

confidence: 99%

Revisiting G-Dedekind domains

Zafrullah¹

2021

Preprint

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Let R be an integral domain with qf (R) = K and let F (R) be the set of nonzero fractional ideals of R. Call R a dually compact domain (DCD) if for each I ∈ F (R) the ideal Iv = (I −1 ) −1 is a finite intersection of principal fractional ideals. We characterize DCDs and show that the class of DCDs properly contains various classes of integral domains, such as Noetherian, Mori and Krull domains. In addition we show that a Schreier DCD is a GCD domain with the property that for each A ∈ F (R) the ideal Av is principal. We show that a domain R is G-Dedekind domain (i.e. has the property that Av is invertible for each A ∈ F (R)) if and only if R is a DCD satisfying the property * : for all pairs of subsets {a 1 , ..., am}, {b 1 , ...bn} ⊆ K\{0},We discuss what the appropriate name for G-Dedekind domains and related notions should be. We also make some observations about how the DCDs behave under localizations and polynomial ring extensions.

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Section: Of Course By Saying That An Idealmentioning

confidence: 99%

Revisiting G-Dedekind domains

Zafrullah¹

2021

Preprint

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“…We denote the set of semistar operations by SStar(D). For the standard results on the theory of star and semistar operations, the reader may consult [8, Chapter 32], [16] or [3].…”

Section: Notation and Backgroundmentioning

confidence: 99%

“…Indeed, star and semistar operations are often introduced and studied together (and the latter were actually born as a generalization of the former [16]); on the other hand, restricting a semistar operation we get a semiprime operation, and this correspondence can partly be inverted [5]. However, the study of these classes is usually pursued in different contexts: star and semistar operations are defined only in the integral domain setting (although they can be generalized: see [3]), and their study is often connected with Prüfer domain and their generalizations, while semiprime operations can be defined for arbitrary rings, and are studied especially in the Noetherian context. In particular, a major point of difference is that the most useful semiprime operations have some functorial properties, while star and semistar operations usually behave very badly under quotients, with only some sparse exception [7].…”

Section: Introductionmentioning

confidence: 99%

Multiplicative closure operations on ring extensions

Spirito¹

2019

Preprint

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Let A ⊆ B be a ring extension and G be a set of Asubmodules of B. We introduce a class of closure operations on G (which we call multiplicative operations on (A, B, G)) that generalizes the classes of star, semistar and semiprime operations. We study how the set Mult(A, B, G) of these closure operations vary when A, B or G vary, and how Mult(A, B, G) behave under ring homomorphisms. As an application, we show how to reduce the study of star operations on analytically unramified one-dimensional Noetherian domains to the study of closures on finite extensions of Artinian rings.

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“…and is called the t-operation on D. These are examples of the so called star operations. The reader may consult Jesse Elliott's book [20] for these operations. A fractional ideal I is called a v-ideal (resp.…”

Section: Introductionmentioning

confidence: 99%

Semirigid GCD domains II

Zafrullah

2021

J. Algebra Appl.

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Let [Formula: see text] be an integral domain with quotient field [Formula: see text] throughout[Formula: see text] Call two elements [Formula: see text][Formula: see text]-coprime if [Formula: see text] Call a nonzero non-unit [Formula: see text] of an integral domain [Formula: see text] rigid if for all [Formula: see text] we have [Formula: see text] or [Formula: see text] Also, call [Formula: see text] semirigid if every nonzero non-unit of [Formula: see text] is expressible as a finite product of rigid elements. We show that a semirigid domain [Formula: see text] is a GCD domain if and only if [Formula: see text] satisfies [Formula: see text] product of every pair of non-[Formula: see text]-coprime rigid elements is again rigid. Next, call [Formula: see text] a valuation element if [Formula: see text] for some valuation ring [Formula: see text] with [Formula: see text] and call [Formula: see text] a VFD if every nonzero non-unit of [Formula: see text] is a finite product of valuation elements. It turns out that a valuation element is what we call a packed element: a rigid element [Formula: see text] all of whose powers are rigid and [Formula: see text] is a prime ideal. Calling [Formula: see text] a semi-packed domain (SPD) if every nonzero non-unit of [Formula: see text] is a finite product of packed elements, we study SPDs and explore situations in which a variant of an SPD is a semirigid GCD domain.

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Rings, Modules, and Closure Operations

Cited by 27 publications

References 0 publications

Revisiting G-Dedekind domains

Revisiting G-Dedekind domains

Multiplicative closure operations on ring extensions

Semirigid GCD domains II

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