Let $$R$$ be an associative ring with identity and $$N$$ a (right) $$R$$-module. $$N$$ is said to be a Whitehead test module for projectivity (or a p-test module) when $$\text{Ext}_R(M,N)=0$$ implies that $$M$$ is projective for each $$R$$-module $$M$$. The concept of Whitehead test module for injectivity (i-test module) is defined dually by the condition that $$\text{Ext}_R(N,M)=0$$ implies $$M$$ injective. The problem of deciding whether a given module is p-test can be very hard. For example, the question is $$\mathbb{Z}$$ a p-test $$\mathbb{Z}$$-module? is exactly the Whitehead problem, which by results of S. Shelah is known to be undecidable in ZFC+GCH (Zermelo-Fraenkel set theory plus the Generalized Continuum Hypothesis).
The last part of the paper is devoted to the study of rings with many test modules. If $$R$$ is not semisimple, then no projective module is i-test and so the author approaches the problem by looking at when the class of i-test modules contains all the non-projective modules of small size. Thus a non-semisimple ring $$R$$ is said to be fully saturated ($$\kappa$$-saturated) when all non-projective (resp. $$\leq\kappa$$-generated non-projective) modules are i-test. Fully saturated rings had already been studied by the author under the name of ‘right Ext-rings’ [Associative rings and the Whitehead property of modules (1990; Zbl 0692.16017)] and here the results of the preceding monograph are substantially improved. The author shows that the classification of saturated rings can be reduced to the case of the indecomposable ones and these, in turn, fall into two classes when they are 1-saturated: type I, if all simple modules are isomorphic, and type II, the rest. The rings of type II are characterized as certain generalized upper triangular matrix rings and, among other interesting results, it is also proved that it is consistent that each fully saturated ring of type I is a full matrix ring over a local quasi-Frobenius ring. The results of the paper have been applied by the author and R. Colpi to the study of tilting theory and of *-modules.