Whitehead test modules

A (right $R$-) module $N$ is said to be a Whitehead test module for projectivity (shortly: a p-test module) provided for each module $M$, $Ext_R(M,N)=0$ implies $M$ is projective. Dually, i-test modules are defined. For example, $\mathbb{Z}$ 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 ($\kappa$-saturated) provided that all non-projective ($\le\kappa$-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 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 $\ast$-modules.