Whitehead test modules

Abstract

A (right R-) module N is said to be a Whitehead test module for projectivity (shortly: a p-test module) provided for each module M , ExtR(M,N) = 0 implies M is projective. Dually, i-test modules are defined. For example, Z is a p-test abelian group iff each Whitehead group is free. Our first main result says that if R is a right hereditary non-right perfect ring, then the existence of p-test modules is independent of ZFC + GCH. On the other hand, for any ring R, there is a proper class of i-test modules. Dually, there is a proper class of p-test modules over any right perfect ring. A non-semisimple ring R is said to be fully saturated (κ-saturated) provided that all non-projective (≤ κ-generated non-projective) modules are i-test. We show that classification of saturated rings can be reduced to the indecomposable ones. Indecomposable 1-saturated rings fall into two classes: type I, where all simple modules are isomorphic, and type II, the others. Our second main result gives a complete characterization of rings of type II as certain generalized upper triangular matrix rings, GT (1, n, p, S, T ). The four parameters involved here are skew-fields S and T , and natural numbers n, p. For rings of type I, we have several partial results: e.g. using a generalization of Bongartz Lemma, we show that it is consistent that each fully saturated ring of type I is a full matrix ring over a local quasi-Frobenius ring. In several recent papers, our results have been applied to Tilting Theory and to the Theory of ∗-modules. In modern algebra, the structure of rings, R, is studied by means of properties of corresponding module categories, Mod-R. In most cases, it is not possible to characterize Mod-R fully. Nevertheless, there are important subclasses of Mod-R that can be treated in detail and that shed light on the whole of Mod-R. Among the prominent ones are the classes of all projective and all injective modules. Recall that a module M is said to be projective (injective) provided that the functorHomR(M,−) (HomR(−,M)) preserves short exact sequences. There is also a universal algebraic aspect: each module is a factor module of a projective module, and a submodule of an injective module. So a possible strategy to investigate ModR consists in describing all injective modules, and for each injective module, I, all its submodules. The first step is usually relatively easy, but the second may be quite hard. For example, using this strategy for abelian groups, one meets serious difficulties already for I = Q⊕Q (see e.g. [E, Theorem 2]). Received by the editors March 17, 1995. 1991 Mathematics Subject Classification. Primary 16E30; Secondary 03E35, 20K35.