Ext and von Neumann regular rings. (English) Zbl 0577.16007

A ring R is called a left T-ring if \(Ext_ R(M,N)\neq 0\) for each non- projective module M and each non-injective module N. In the paper, the following results are proved: 1) Let R be a von Neumann regular ring. If R is a T-ring, then each left ideal of R is countably generated. 2) Let R be a simple countable regular ring. Then \(Ext_ R(M,N)\neq 0\) for all countably generated modules M, N such that M is not projective and N is not injective. 3) The preceding result is also true for uncountably generated non-projective modules M if we work in the ZFC set theory plus the Axiom of constructibility. 4) Let R be a direct limit of a countable directed system of simple countable completely reducible rings. Then R is a simple countable regular ring and R is not a T-ring, provided R is not completely reducible.
Reviewer: T.Kepka


16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16Exx Homological methods in associative algebras
16S20 Centralizing and normalizing extensions
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
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[1] F. W. Anderson, K. R. Fuller: Rings and categories of modules. Springer-Verlag, New York-Heidslberg-Berlin, 1974. · Zbl 0301.16001
[2] L. Bican T. Kepka, P. Němec: Rings, modules, and preradicals. M. Dekker Inc., New York-Basel, 1982. · Zbl 0483.16026
[3] H. Cartan, S. Eilenberg: Homological algebra. Princeton University Press, Princeton, 1956. · Zbl 0075.24305
[4] S. E. Dickson: A torsion theory for abelian categories. Trans. Amer. Math. Soc. 121 (1966), 223-235. · Zbl 0138.01801
[5] P. C. Eklof: Homological algebra and set theory. Trans. Amer. Math. Soc. 227 (1977), 207–225. · Zbl 0355.02047
[6] B. J. Gardner: Rings whose modules form few torsion classes. Bull. Austral. Math. Soc. 4 (1971), 355-359. · Zbl 0209.07304
[7] K. R. Goodearl: Von Neumann regular rings. Pitman, London-San Francisco-Melbourne, 1979. · Zbl 0411.16007
[8] P. Jambor: An orthogonal theory of a set-valued bifunctor. Czech. Math. J. 23 (1973), 447-454. · Zbl 0271.18010
[9] P. Jambor: Hereditary tensor-orthogonal theories. Comment. Math. Univ. Carolinae 16 (1975), 139-145. · Zbl 0303.16011
[10] S. Shelah: A compactness theorem for singular cardinals, free algebras. Whitehead problem and transversals. Israel J. Math. 21 (1975), 319-349. · Zbl 0369.02034
[11] J. Trlifaj, T. Kepka: Structure of T-rings. in Radical Theory (Proc. Conf. Eger, 1982), Colloq. Math. Soc. Bolyai, North-Holland, Amsterdam) · Zbl 0588.16021
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