Abstract elementary classes of R-modules with pure submodules are not always stable, but tame ones are almost stable and those with amalgamation are stable assuming a strongly compact cardinal.
Shelah.Categoricity for abstract classes with amalgamation
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On the problem of stability of abstract elementary classes of modules
Abstract elementary classes of R-modules with pure submodules are not always stable, but tame ones are almost stable and those with amalgamation are stable assuming a strongly compact cardinal.