Appendix D - Model theory of modules: an update

Summary

For most of the model theory of modules per se, and certainly for model-theoretic stability theory in the context of modules, my earlier book, [495], rather than this one, is the place to look, although for a model-theoretic view of some more recent developments one may look at the later parts of the paper [516]. At the time [495] was written it seemed that all the basic results were in place and most of the general development had been carried through, with a limited number of hard model-theoretic problems still open. This was not correct: elementary duality at the level of theories and Ziegler spectra was still to come. Also, with the internal development seemingly complete, the way forward was clearly to applications and perhaps analogous developments in other contexts. That was largely correct, though the equivalence of the category of imaginaries with the category of finitely presented functors, and related functor-category reformulations was still a major theme to come.

These new features, elementary duality and equivalence with functor categories, have been dealt with in the main body of this text. Progress on the major purely model-theoretic open questions (Vaught's conjecture, various questions still open in the absence of a countability hypothesis) has been significant, in that they have often been settled for particular types of ring, but disappointing in that all the main questions left open in Ziegler's paper [726] still are open in the general case.