On the decidability of the theory of modules over the ring of algebraic integers

L’Innocente, Sonia; Toffalori, Carlo; Puninski, Gena

doi:10.1016/j.apal.2017.02.003

Cited by 6 publications

(12 citation statements)

References 16 publications

(40 reference statements)

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“…The decidability of the theory of modules over the ring of algebraic integers Z has also been obtained by S. L'Innocente, G. Puninski and C. Toffalori, using different methods [19]. 6.2.…”

Section: Applicationsmentioning

confidence: 96%

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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“…The decidability of the theory of modules over the ring of algebraic integers Z has also been obtained by S. L'Innocente, G. Puninski and C. Toffalori, using different methods [19]. 6.2.…”

Section: Applicationsmentioning

confidence: 96%

Bézout domains and lattice-valued modules

L’Innocente

Point

2020

Journal of Pure and Applied Algebra

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“…Each B P is a valuation domain, and the Ziegler spectrum of this class of rings was thoroughly investigated (see [17, Chapters 12,13], or [6] for recent development). In more detail, let Γ denote the value group of a valuation domain V .…”

Section: Weakly Prime Idealsmentioning

confidence: 99%

“…However, as it was mentioned there, this information is expected to be elaborated for particular classes of Bezout domains. One example of this refinement was given in [22], and some information on the structure of the Ziegler spectrum of the ring of algebraic integers is contained in a recent preprint [16].…”

Section: Introductionmentioning

confidence: 99%

The Ziegler Spectrum of the Ring of Entire Complex Valued Functions

L’Innocente¹,

Point²,

Puninski

et al. 2019

J. symb. log.

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“…Proof. By [10,Lemma 3.3] (using the Krull dimension 1 hypothesis) the prime radical relation a ∈ rad(bR) can be decided effectively for a, b ∈ R.…”

Section: N) Are Pairwise Comparable Under Inclusionmentioning

confidence: 99%

“…The theory of modules of Bézout domains of the form D + XQ[X] ⊆ Q[X], where D is a principal ideal domain with field of fractions Q, is shown in [PT14] to be decidable under certain reasonable effective conditions on D. In particular, it is shown that Z + XQ[X] has decidable theory of modules. The theory of modules of the ring of algebraic integers, along with some other Bézout domains with Krull dimension 1, is shown to have decidable theory of modules in [LTP17].…”

Section: Introductionmentioning

confidence: 99%

Decidability of the Theory of Modules Over Prüfer Domains With Infinite Residue Fields

Gregory¹,

L’Innocente²,

Puninski

et al. 2018

J. symb. log.

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We provide algebraic conditions ensuring the decidability of the theory of modules over effectively given Prüfer (in particular Bézout) domains with infinite residue fields in terms of a suitable generalization of the prime radical relation. For Bézout domains these conditions are also necessary. IntroductionWe deal here with decidability of first order theories of modules over Prüfer (in particular Bézout) domains R with infinite residue fields. We assume R effectively given (so countable), in order to ensure that the decision problem for R-modules makes sense.The model theory of modules over Bézout domains, with some hints at Prüfer domains, is studied in [16]. The decidability of the theory of modules over the ring of algebraic integers is proved in [10] (see also [8]), and a similar result is obtained in [15] over Bézout domains obtained from principal ideal domains by the so called D+M-construction [3].On the other hand Gregory [5], extending [14], proved that the theory of modules over a(n effectively given) valuation domain V is decidable if and only if there is an algorithm which decides the prime radical relation, namely, for every a, b ∈ V , answers whether a ∈ rad(bV ) (equivalently, whether the prime ideals of V containing b also include a).This paper develops a similar analysis in a closely related setting, that is, over Prüfer domains.In fact a domain is Prüfer if and only if all its localizations at maximal ideals are valuation domains. Bézout domains are a notable subclass of Prüfer domains. In both cases we focus on the domains all of whose residue fields with respect to maximal ideals are infinite. The reason and the benefit of this choice are illustrated in § 3 below. Notice that Prüfer (indeed Bézout) domains with infinite residue fields include the ring of algebraic integers and the ring of complex valued entire functions -even if the latter is uncountable and so cannot be effectively given (but see the analysis of its Ziegler spectrum in [9]). Other noteworthy examples will be proposed in § 6.2000 Mathematics Subject Classification. 03C60 (primary), 03C98, 03B25, 13F05. while this paper was being completed. The other three authors would like to dedicate this article as a tribute to his memory. 1Our main result, resembling [5], states that, if R is such a Bézout domain, then the theory of R-modules is decidable if and only if there is an algorithm which answers a sort of double prime radical relation, in detail, given a, b, c, d ∈ R, decides whether, for all prime ideals p, q with p+q = R, b ∈ p implies a ∈ p or d ∈ q implies c ∈ q. This will be proved in § 6. Generalizations to Prüfer domains will be presented in the final part of the paper, in § 7. The preceding sections § § 2-5 describe the framework of (effectively given) Prüfer domains and prepare the main theorems.We refer to all the already mentioned papers and books, as well as to the key references on model theory of modules, [11], [12] and [18]. We also assume some familiarity with Prüfer domains, as treated, for instance, in [3] an...

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On the decidability of the theory of modules over the ring of algebraic integers

Abstract: Abstract. We will prove that the theory of all modules over the ring of algebraic integers is decidable.

Cited by 6 publications

References 16 publications

Bézout domains and lattice-valued modules

Bézout domains and lattice-valued modules

The Ziegler Spectrum of the Ring of Entire Complex Valued Functions

Decidability of the Theory of Modules Over Prüfer Domains With Infinite Residue Fields

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