This chapter is devoted to the model theory of projective and, more generally, flat modules. As with the dual case of injective and absolutely pure modules (Chapter 15), one obtains relatively “complete” results. In both cases, the key step is the description of the particular form taken by the pp-definable subgroups.

If a module M is flat, then every pp-definable subgroup of it has the form φ(M) = M.φ(RR): indeed, this property characterises the flat modules. It follows that the model-theoretic complexity of a flat module can be no greater than that of the ring. We see (§1) that, if the ring is left coherent, then its pp-definable subgroups are precisely the finitely generated left ideals. We deduce that the class of flat modules is axiomatisable iff the ring is left coherent.

It follows from the results of §1 that a ring which is left coherent is totally transcendental as a module over itself iff it is right perfect: but we do not have a general algebraic characterisation of the totally transcendental rings. We then note that the left coherent, right perfect rings are precisely those over which the class of projective modules is elementary (over such a ring, every flat module is projective). The section finishes with a characterisation of those rings over which the free modules form an elementary class.

Definable subgroups of flat and projectiue modules

A major step in understanding the model theory of any particular class of structures is the characterisation of the definable sets. It is shown below that if M is a flat module and if ip is a pp formula, then φ(M) = M.φR.