83 research outputs found
â”´N as an Abstract Elementary Class
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Relative injective modules, superstability and noetherian categories
We study classes of modules closed under direct sums,
-submodules and -epimorphic images where
is either the class of embeddings, -embeddings or pure
embeddings.
We show that the -injective modules of theses classes satisfy a
Baer-like criterion. In particular, injective modules, -injective modules,
pure injective modules, flat cotorsion modules and -torsion pure
injective modules satisfy this criterion. The argument presented is a model
theoretic one. We use in an essential way stable independence relations which
generalize Shelah's non-forking to abstract elementary classes.
We show that the classical model theoretic notion of superstability is
equivalent to the algebraic notion of a noetherian category for these classes.
We use this equivalence to characterize noetherian rings, pure semisimple
rings, perfect rings and finite products of finite rings and artinian valuation
rings via superstability.Comment: 25 page
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