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.