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On weakly regular rings and generalizations of V-rings. (English) Zbl 1253.16010

Summary: We study weakly regular rings and some generalizations of V-rings via GW-ideals. We show that: (1) If \(R\) is a left weakly regular ring whose maximal left (right) ideals are GW-ideals, then \(R\) is strongly regular; (2) If \(R\) is a right weakly regular ring whose maximal essential left ideals are GW-ideals, then \(R\) is ELT regular; (3) If \(R\) is a ring in which \(l(a)\) is a GW-ideal for all \(a\in R\), then \(R\) is left weakly regular if and only if \(R\) is right weakly regular; (4) A ring \(R\) is strongly regular if and only if \(R\) is a ZI left GP-V\('\)-ring whose maximal essential left (right) ideals are GW-ideals; (5) If \(R\) is a left (right) GP-V-ring such that \(l(a)\) is a GW-ideal for all \(a\in R\), then \(R\) is weakly regular.

MSC:

16E50 von Neumann regular rings and generalizations (associative algebraic aspects)
16D25 Ideals in associative algebras
16D80 Other classes of modules and ideals in associative algebras