WHITEHEAD TEST MODULES

A (right R-) module N is said to be a Whitehead test module for projec tivity (shortly: a p-test module) provided for each module M, Ext(R)(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 f ree. Our first main result says that if R is a right hereditary non-ri ght 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 cla ss of i-test modules. Dually, there is a proper class of p-test module s over any right perfect ring. A non-semisimple ring R is said to be f ully saturated (kappa-saturated) provided that all non-projective (les s than or equal to kappa-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 cl asses: type I, where all simple modules are isomorphic, and type II, t he 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 sever al 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 rece nt papers, our results have been applied to Tilting Theory and to the Theory of -modules.