Submission date: 6 October 2020

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), ≤_p) is stable. In [Maz4, 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. 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, ≤_p) is λ-stable for every λ such that λ^{|R| + ℵ_0}= λ.
- If K is closed under pure submodules and pure epimorphic images, then (K, ≤_p) is λ-stable for every λ such that λ^{|R| + ℵ_0}= λ.
- Assume R is left Von Neumann regular. If K is closed under submodules and has arbitrarily large models, then (K, ≤_p) 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, 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.

Mathematics Subject Classification: Primary: 03C48 Secondary: 03C45, 03C60, 13L05, 16D10, 16P40

Keywords and phrases:

Full text arXiv 2010.02918: pdf, ps.