Abstract Fisher [10] and Baur [6] 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 [25, 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: 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 pure-injective envelopes, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda \geq \operatorname {LS}(\mathbf {K})$ such that $\lambda ^{|R| + \aleph _0}= \lambda $ . • If K is closed under pure submodules and pure epimorphic images, then $\mathbf {K}$ is $\lambda $ -stable for every $\lambda $ such that $\lambda ^{|R| + \aleph _0}= \lambda $ . • Assume R is Von Neumann regular. If $\mathbf {K}$ is closed under submodules and has arbitrarily large models, then $\mathbf {K}$ 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 $\mathfrak {s}$ -torsion modules.

We introduce and study simple and supersimple independence relations in the context of AECs with a monster model. Theorem 0.1Let K be an AEC with a monster model.•If K has a simple independence relation, then K does not have the 2-tree property.•If K has a simple independence relation with the (<ℵ0)-witness property for singletons, then K does not have the tree property. Theorem 0.1 Let K be an AEC with a monster model. If K has a simple independence relation, then K does not have the 2-tree property. If K has a simple independence relation with the (<ℵ0)-witness property for singletons, then K does not have the tree property. The proof of both facts is done by finding cardinal bounds to classes of small Galois-types over a fixed model that are inconsistent for large subsets. We think that this finer way of counting types is an interesting notion in itself. We characterize supersimple independence relations by finiteness of the Lascar rank under locality assumptions on the independence relation.

We uncover a connection between the model-theoretic notion of superstability and that of noetherian rings and pure-semisimple rings. We characterize noetherian rings via superstability of the class of left modules with embeddings. Theorem 0.1For a ring R the following are equivalent.(1)R is left noetherian.(2)The class of left R-modules with embeddings is superstable.(3)For every λ≥|R|+ℵ0, there is χ≥λ such that the class of left R-modules with embeddings has uniqueness of limit models of cardinality χ.(4)Every limit model in the class of left R-modules with embeddings is Σ-injective. Theorem 0.1 For a ring R the following are equivalent. R is left noetherian. The class of left R-modules with embeddings is superstable. For every λ≥|R|+ℵ0, there is χ≥λ such that the class of left R-modules with embeddings has uniqueness of limit models of cardinality χ. Every limit model in the class of left R-modules with embeddings is Σ-injective. We characterize left pure-semisimple rings via superstability of the class of left modules with pure embeddings. Theorem 0.2For a ring R the following are equivalent.(1)R is left pure-semisimple.(2)The class of left R-modules with pure embeddings is superstable.(3)There exists λ≥(|R|+ℵ0)+ such that the class of left R-modules with pure embeddings has uniqueness of limit models of cardinality λ.(4)Every limit model in the class of left R-modules with pure embeddings is Σ-pure-injective. Theorem 0.2 For a ring R the following are equivalent. R is left pure-semisimple. The class of left R-modules with pure embeddings is superstable. There exists λ≥(|R|+ℵ0)+ such that the class of left R-modules with pure embeddings has uniqueness of limit models of cardinality λ. Every limit model in the class of left R-modules with pure embeddings is Σ-pure-injective. Both equivalences provide evidence that the notion of superstability could shed light in the understanding of algebraic concepts. As this paper is aimed at model theorists and algebraists an effort was made to provide the background for both.

Abstract Shelah has provided sufficient conditions for an ${\Bbb L}_{\omega _1 ,\omega } $ -sentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ${\aleph _n}$ ’s. Using tools of Boney, Shelah, and the second author, we give assumptions on ${\aleph _0}$ and ${\aleph _1}$ which suffice when ψ is restricted to be universal: Theorem. Assume ${2^{{\aleph _0}}} < {2^{{\aleph _1}}}$ . Let ψ be a universal ${\Bbb L}_{\omega _1 ,\omega } $ -sentence. (1) If ψ is categorical in ${\aleph _0}$ and $1 \leqslant {\Bbb L}\left( {\psi ,\aleph _1 } \right) < 2^{\aleph _1 } $ , then ψ has arbitrarily large models and categoricity of ψ in some uncountable cardinal implies categoricity of ψ in all uncountable cardinals. (2) If ψ is categorical in ${\aleph _1}$ , then ψ is categorical in all uncountable cardinals. The theorem generalizes to the framework of ${\Bbb L}_{\omega _1 ,\omega } $ -definable tame abstract elementary classes with primes.