Elimination of quantifiers

Abstract

A complete theory of modules has (complete) elimination of quantifiers if every definable subset of a model may be defined without the use of quantifiers. Thus, in a module with elimination of quantifiers, the (pp-)type of every element is determined by its annihilator – so our study comes close to being “purely algebraic”. A good deal of initial work in the model theory of modules was done in a context where one has elimination of quantifiers, and that comparatively “algebraic” case has proved to be quite a reliable guide as to what to expect when we do have to take account of quantifiers. The first section begins by delineating some of the consequences of elimination of quantifiers. One soon discovers that elimination of quantifiers is just a little weaker than one would like: so we introduce and work with the stronger condition (denoted elim-Q + ) that every pp formula is actually a conjunction of atomic formulas (elimination of quantifiers guarantees only that it is a boolean combination of atomic formulas). In any theory with elim-Q + the indecomposable pureinjectives are “small” in the sense that they are uniform (any two non-zero submodules intersect non-trivially). A ring is regular iff all of its modules have complete elimination of quantifiers. That is proved in the second section, and the consequences are pursued. The usual spectrum of algebraic finiteness conditions is much shortened for regular rings: we see that the same goes for our model-theoretic finiteness conditions. All our work on regular rings is aided by the fact that there is a simple canonical form for the invariants of §2.4.