# Some stable non-elementary classes of modules

Research paper by **Marcos Mazari-Armida**

Indexed on: **07 Oct '20**Published on: **06 Oct '20**Published in: **arXiv - Mathematics - Logic**

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#### Abstract

Fisher [Fis75] and Baur [Bau75] showed independently in the seventies that
for any $T$ a complete first-order theory extending the theory of modules,
$(Mod(T), \leq_p)$ is stable. In [Maz4, 2.12], it is asked if the same is true
for any abstract elementary class $(K, \leq_p)$ such that $K$ is a class of
modules and $\leq_p$ is the pure submodule relation. In this paper we give some
instances where this is true:
$\textbf{Theorem.}$ Assume $R$ is an associative ring with unity. Let $(K,
\leq_p)$ be an AEC such that $K \subseteq R\text{-Mod}$ and $K$ is closed under
finite direct sums, then:
- If $K$ is closed under direct summands and pure-injective envelopes, then
$(K, \leq_p)$ is $\lambda$-stable for every $\lambda$ such that $\lambda^{|R| +
\aleph_0}= \lambda$.
- If $K$ is closed under pure submodules and pure epimorphic images, then
$(K, \leq_p)$ is $\lambda$-stable for every $\lambda$ such that $\lambda^{|R| +
\aleph_0}= \lambda$.
- Assume $R$ is left Von Neumann regular. If $K$ is closed under submodules
and has arbitrarily large models, then $(K, \leq_p)$ is $\lambda$-stable for
every $\lambda$ such that $\lambda^{|R| + \aleph_0}= \lambda$.
As an application of these results we give new characterizations of
noetherian rings, pure-semisimple 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.