| Abstract: | A (right  -) module  is said to be a Whitehead test module for projectivity (shortly: a p-test module) provided for each module  ,  implies is projective. Dually, i-test modules are defined. For example,  is a p-test abelian group iff each Whitehead group is free. Our first main result says that if 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 , 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 is said to be fully saturated ( -saturated) provided that all non-projective ( -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,  . The four parameters involved here are skew-fields  and  , and natural numbers  . 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. |
