Abelian lattice-ordered groups and a characterization of the maximal spectrum of a Prüfer domain

Rump, Wolfgang

doi:10.1016/j.jpaa.2014.03.011

Cited by 5 publications

(2 citation statements)

References 17 publications

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“…Then Γ is projectable if for any g ∈ Γ, g ⊥ is a cardinal summand [12,Section 3.5]. Since MSpec(B) is homeomorphic to Min(Γ(B)) [32,Proposition 8], the property for B to have good factorization, translates into the property for Γ(B) to be a projectable ℓ-group. Remark 5.5.…”

Section: Localizationsmentioning

confidence: 99%

Bézout domains and lattice-valued modules

L’Innocente

Point

2020

Journal of Pure and Applied Algebra

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Let B be a commutative Bézout domain B and let M Spec(B) be the maximal spectrum of B. We obtain a Feferman-Vaught type theorem for the class M od B of B-modules. We analyse the definable sets in terms, on one hand, of the definable sets in the classes M od BM , where B M ranges over the localizations of B at M ∈ M Spec(B), and on the other hand, of the constructible subsets of M Spec(B). When B has good factorization, it allows us to derive decidability results for the class M od B , in particular when B is the ring Z of algebraic integers or the one of rings Z ∩ R, Z ∩ Q p .MSC 2010 classification: 03C60, 03B25, 13A18, 06F15.

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Section: Localizationsmentioning

confidence: 99%

Bézout domains and lattice-valued modules

L’Innocente

Point

2020

Journal of Pure and Applied Algebra

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“…In pointfree topology, z-ideals were introduced by Dube in [8] in terms of the cozero map. z-Ideals have been studied in the theory of abelian lattice-ordered groups [4,27] and in the context of Riesz space in [17] and [18].…”

Section: Introductionmentioning

confidence: 99%

zc-ideals and prime ideals in the ring RcL

Estaji¹,

Feizabadi²,

Sarpoushi³

2018

Filomat

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The ring R c L is introduced as a sub-f-ring of RL as a pointfree analogue to the subring C c (X) of C(X) consisting of elements with the countable image. We introduce z c-ideals in R c L and study their properties. We prove that for any frame L, there exists a space X such that βL OX with C c (X) R c (OX) R c βL R * c L, and from this, we conclude that if α, β ∈ R c L, |α| ≤ |β| q for some q > 1, then α is a multiple of β in R c L. Also, we show that IJ = I ∩ J whenever I and J are z c-ideals. In particular, we prove that an ideal of R c L is a z c-ideal if and only if it is a z-ideals. In addition, we study the relation between z c-ideals and prime ideals in R c L. Finally, we prove that R c L is a Gelfand ring.

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Elementary divisors, Hochster duality, and spectra

Rump

2024

Arch. Math.

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Abelian lattice-ordered groups and a characterization of the maximal spectrum of a Prüfer domain

Cited by 5 publications

References 17 publications

Bézout domains and lattice-valued modules

Bézout domains and lattice-valued modules

zc-ideals and prime ideals in the ring RcL

Elementary divisors, Hochster duality, and spectra

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