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On superstability in the class of flat modules and perfect rings


Author: Marcos Mazari-Armida
Journal: Proc. Amer. Math. Soc. 149 (2021), 2639-2654
MSC (2020): Primary 03C48, 16B70; Secondary 03C45, 03C60, 13L05, 16L30, 16D10
DOI: https://doi.org/10.1090/proc/15359
Published electronically: March 22, 2021
MathSciNet review: 4246813
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Abstract:

We obtain a characterization of left perfect rings via superstability of the class of flat left modules with pure embeddings.

\begingroup\itshapeTheorem 0.1. For a ring $R$ the following are equivalent.

  1. $R$ is left perfect.

  2. The class of flat left $R$-modules with pure embeddings is superstable.

  3. There exists a $\lambda \geq (|R| + \aleph _0)^+$ such that the class of flat left $R$-modules with pure embeddings has uniqueness of limit models of cardinality $\lambda$.

  4. Every limit model in the class of flat left $R$-modules with pure embeddings is $\Sigma$-cotorsion.

\endgroup

A key step in our argument is the study of limit models in the class of flat modules. We show that limit models with chains of long cofinality are cotorsion and that limit models are elementarily equivalent.

We obtain a new characterization via limit models of the rings characterized in Rothmaler [Comm. Algebra 30 (2002), pp. 3077–3085]. We show that in these rings the equivalence between left perfect rings and superstability can be refined. We show that the results for these rings can be used to extend Shelah result [1.2, Notre Dame J. Form. Log. 58 (2017), pp. 159–177] to classes of flat modules not axiomatizable in first-order logic.


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Additional Information

Marcos Mazari-Armida
Affiliation: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213
MR Author ID: 1301661
ORCID: 0000-0002-5302-671X
Email: mmazaria@andrew.cmu.edu

Keywords: Superstability, perfect rings, limit models, cotorsion modules, flat modules, abstract elementary classes
Received by editor(s): October 25, 2019
Received by editor(s) in revised form: September 8, 2020, and October 6, 2020
Published electronically: March 22, 2021
Communicated by: Heike Mildenberger
Article copyright: © Copyright 2021 American Mathematical Society