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

Trlifaj, Jan

Czech. Math. J. 35(110), 324-332 (1985).

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

MSC:

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)

Keywords:

left T-ring; von Neumann regular ring; simple countable regular ring; countably generated modules; ZFC; Axiom of constructibility; completely reducible rings

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