SOME STABLE NON-ELEMENTARY CLASSES OF MODULES

Abstract Fisher [10] and Baur [6] showed independently in the seventies that if T is a complete first-order theory extending of modules, then class models with pure embeddings stable. In [25, 2.12], it asked same true for any abstract elementary $(K, \leq _p)$ such K modules $\leq _p$ submodule relation. this paper we give some instances where true: Theorem. Assume R an associative ring unity. Let be AEC $K \subseteq R\text {-Mod}$ closed under finite direct sums, then: • If pure-injective envelopes, $\mathbf {K}$ $\lambda $ -stable every \geq \operatorname {LS}(\mathbf {K})$ ^{|R| + \aleph _0}= \lambda . submodules epimorphic images, Von Neumann regular. has arbitrarily large models, As application these results new characterizations noetherian rings, pure-semisimple Dedekind domains, fields via superstability. Moreover, show how can used to link between being good stability hierarchy axiomatizability hierarchy. Another existence universal respect several classes modules. Among them, flat $\mathfrak {s}$ -torsion