Whitehead test modules
Author: Jan Trlifaj
Journal: Trans. Amer. Math. Soc. 348 (1996), 1521-1554
MSC (1991): Primary 16E30; Secondary 03E35, 20K35
MathSciNet review: 1322958
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.
Affiliation: address Department of Algebra, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 00 Prague 8, The Czech Republic
Keywords: Whitehead test module, injective module, projective module, uniformization, weak diamond, generalized upper triangular matrix rings
Additional Notes: Research supported by grant GAUK-44.
Article copyright: © Copyright 1996 American Mathematical Society