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**Whitehead test modules.**
*(English)*
Zbl 0865.16006

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 author investigates the existence of p-test and i-test modules and shows that the i-test modules form a proper class for an arbitrary ring and, dually, the p-test class is a proper class for right perfect rings. The problem of the existence of p-test modules over non-right perfect rings is attacked using the set-theoretic methods developed by Shelah for the Whitehead problem. In particular, the consistency with ZFC+GCH of the non-existence of p-test modules is proved using a uniformization principle due to Shelah. Also, the consistency of the existence of a proper class of p-test modules for any right hereditary non-right perfect ring is proved applying a generalized weak diamond principle.

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.

The author investigates the existence of p-test and i-test modules and shows that the i-test modules form a proper class for an arbitrary ring and, dually, the p-test class is a proper class for right perfect rings. The problem of the existence of p-test modules over non-right perfect rings is attacked using the set-theoretic methods developed by Shelah for the Whitehead problem. In particular, the consistency with ZFC+GCH of the non-existence of p-test modules is proved using a uniformization principle due to Shelah. Also, the consistency of the existence of a proper class of p-test modules for any right hereditary non-right perfect ring is proved applying a generalized weak diamond principle.

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.

Reviewer: J.L.Gómez-Pardo (Santiago de Compostela)

### MSC:

16E30 | Homological functors on modules (Tor, Ext, etc.) in associative algebras |

03E55 | Large cardinals |

16B70 | Applications of logic in associative algebras |

16D40 | Free, projective, and flat modules and ideals in associative algebras |

16D50 | Injective modules, self-injective associative rings |

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

20K35 | Extensions of abelian groups |