On superstability in the class of flat modules and perfect rings

Mazari-Armida, Marcos

doi:10.1090/proc/15359

Cited by 11 publications

(26 citation statements)

References 39 publications

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“…The study of limit models for the class of s-torsion modules and the characterization of superstability we obtain parallels that of previous results, [Maz21b], [Maz1] and [Maz2], with the added difficulty that we have to deal with relative pure-injective modules instead of with pure-injective or cotorsion modules. More precisely, we obtain the following result.…”

Section: Introductionsupporting

confidence: 76%

“…Previous results that characterised superstability in classes of modules always corresponded to classical rings [Maz21b], [Maz1], [Maz2]. In this case we do not know if that is the case.…”

Section: S-torsion Modules As An Aecmentioning

confidence: 76%

“…This paper is part of a program to understand AECs of modules: [Maz20], [KuMa20], [Maz21b], [Maz1], [Maz21a], [Maz2]. Other papers that have studied AECs of modules include: [BCG+],…”

Section: Introductionmentioning

confidence: 99%

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A note on torsion modules with pure embeddings

Mazari-Armida¹

2021

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We study Martsinkovsky-Russell torsion modules [MaRu20] with pure embeddings as an abstract elementary class. We give a model-theoretic characterization of the pureinjective and the Σ-pure-injective modules relative to the class of torsion modules assuming that the ring is right semihereditary. Our characterization of relative Σ-pure-injective modules extends the classical charactetization of [GrJe76] and [Zim77, 3.6].We study the limit models of the class and determine when the class is superstable assuming that the ring is right semihereditary. As a corollary, we show that the class of torsion abelian groups with pure embeddings is strictly stable, i.e., stable not superstable.

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Section: Introductionsupporting

confidence: 76%

“…Previous results that characterised superstability in classes of modules always corresponded to classical rings [Maz21b], [Maz1], [Maz2]. In this case we do not know if that is the case.…”

Section: S-torsion Modules As An Aecmentioning

confidence: 76%

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A note on torsion modules with pure embeddings

Mazari-Armida¹

2021

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“…This result can be used to construct universal models with respect to pure embeddings. In particular, we obtain the next result which extends [She17, 1.2], [Maz2,4.6] and [Maz21,3.7].…”

Section: Introductionsupporting

confidence: 75%

“…The result is similar to [KuMa,3.14], but the argument given there cannot be applied in this setting. As the argument given in [Maz2,4.4] works in the more general setting of classes satisfying Hypothesis 3.1 we only sketch the proof.…”

Section: Classes Closed Under Pure-injective Envelopesmentioning

confidence: 99%

Some stable non-elementary classes of modules

Mazari-Armida¹

2020

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Fisher [Fis75] and Baur [Bau75] showed independently in the seventies that if T is a complete first-order theory extending the theory of modules, then the class of models of T with pure embeddings is stable. In [Maz21, 2.12], it is asked if the same is true for any abstract elementary class (K, ≤p) such that K is a class of modules and ≤p is the pure submodule relation. In this paper we give some instances where this is true: Theorem 0.1. Assume R is an associative ring with unity. Let (K, ≤p) be an AEC such that K ⊆ R-Mod and K is closed under finite direct sums, then:• If K is closed under direct summands and pure-injective envelopes, then K is λ-stable for every λ ≥ LS(K) such that λ |R|+ℵ 0 = λ. • If K is closed under pure submodules and pure epimorphic images, then K is λ-stable for every λ such that λ |R|+ℵ 0 = λ. • Assume R is Von Neumann regular. If K is closed under submodules and has arbitrarily large models, then K is λ-stable for every λ such that λ |R|+ℵ 0 = λ.As an application of these results we give new characterizations of noetherian rings, puresemisimple rings, dedekind domains and fields via superstability. Moreover, we show how these results can be used to show a link between being good in the stability hierarchy and being good in the axiomatizability hierarchy.Another application is the existence of universal models with respect to pure embeddings in several classes of modules. Among them, the class of flat modules and the class of injective torsion modules..

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Superstability, noetherian rings and pure-semisimple rings

Mazari-Armida

2021

Annals of Pure and Applied Logic

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On superstability in the class of flat modules and perfect rings

Cited by 11 publications

References 39 publications

A note on torsion modules with pure embeddings

A note on torsion modules with pure embeddings

Some stable non-elementary classes of modules

Superstability, noetherian rings and pure-semisimple rings

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