Modules with few types

Abstract

It is widely believed that a structure theory ought to be developed for the countable models of a countable theory with few countable models. It is often the case, however, that many of the consequences of having few countable models already follow from the weaker hypothesis of having few types. It is under this condition that we undertake here a study of theories of modules. Virtually all of the information that we are able to attain about modules with few types stems from the observation (Thm. 1.2) that if T is a small theory of modules and p(x) is a T-consistent pp-type, then H(p), the pure- injective hull of p, can be decomposed into a finite direct sum of indecom- posable modules. This amounts to saying that all pure types in finitely many variables are, in some sense, of finite weight. It is also useful in proving (Cor. 2.10) that if the ring is commutative, then the annihilator of any subset of the module has a primary decomposition. Garavaglia [G] has developed an elegant structure theory for w-stable modules which is strong enough to settle Vaught’s conjecture for that class of modules. In Section 3, we show (Thm. 3.3) that if T is a small U-rank 1 theory of modules over a right noetherian ring, then T is w-stable. Section 4, in turn, is devoted to a proof of a result with Pillay (Thm. 4.2) that a module with few types and a regular generic type is of U-rank 1 if it is not connected-by-finite. I am indebted to Edgar Enochs, Mike Prest, and Philipp Rothmaler for their helpful suggestions and their interest.