On superstability in the class of flat modules and perfect rings

Mazari-Armida, Marcos

doi:10.1090/proc/15359

On superstability in the class of flat modules and perfect rings
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by Marcos Mazari-Armida

Proc. Amer. Math. Soc. 149 (2021), 2639-2654

DOI: https://doi.org/10.1090/proc/15359

Published electronically: March 22, 2021

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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.

$R$ is left perfect.
The class of flat left $R$-modules with pure embeddings is superstable.
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 $\lambda$.
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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