If K is an algebraically closed field then the sets of n-tuples which may be defined by positive quantifier-free formulas are precisely the sub-varieties of affine n-space. The Chevalley-Tarski theorem says that every definable subset of affine n-space is a boolean combination of such sub-varieties (is “constructible”). The point is that the existential quantifiers introduced by projection may be eliminated: one says that algebraically closed fields have (complete) elimination of quantifiers.

For comparison one may consider the theory of groups. Here the definition of a subset may require arbitrarily large numbers of alternations of quantifiers, and there seems to be no hope of understanding the shape of a general definable set.

Modules are definitely closer to algebraically closed fields than to groups in this regard. For modules have a relative elimination of quantifiers: it turns out that every definable subset of a module is a boolean combination of “pp-definable” cosets. A pp-definable coset is simply the projection of the solution set to a (not necessarily homogeneous) system of R-linear equations. Therefore, such a coset is definable by a formula with only existential quantifiers prefixing a conjunction of atomic formulas (a “positive primitive” formula): we say that modules have ppelimination of quantifiers. It is this fact which brings the model-theoretic and algebraic aspects of modules close together.

This description of the definable subsets is the key to the model-theoretic analysis of modules.

The reader should know that the pace of this chapter is rather leisurely so as to accommodate a wide variation in readers' backgrounds. A number of examples are introduced and many of these are developed further in the text.