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2008•Journal of Baoji University of Arts and SciencesRequires access

The regularity of SF'-rings

Xiao Guang-shi

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Abstract

Aim A ring R is a left(right) SF'-ring if every simple singular left(right) R-module is flat.The regularity of SF'-rings is studied.Methods SF'-rings are discussed on condition that every idempotent element is left semi-center and SF'-rings are LANE-ring.Results Two new characterizations of regular rings are given.(1) If R is a left SF'-ring and R/Z(RR) is reduced,then R is strongly regular;(2) R is strongly regular if and only if R is a left SF'-ring whose every idempotent element of R is left semi-center and is LANE-ring.Conclusion The results are helpful for the problem that SF-rings are regular rings or not.

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Aim A ring R is a left(right) SF'-ring if every simple singular left(right) R-module is flat.The regularity of SF'-rings is studied.Methods SF'-rings are discussed on condition that every idempotent element is left semi-center and SF'-rings are LANE-ring.Results Two new characterizations of regular rings are given.(1) If R is a left SF'-ring and R/Z(RR) is reduced,then R is strongly regular;(2) R is strongly regular if and only if R is a left SF'-ring whose every idempotent element of R is left semi-center and is LANE-ring.Conclusion The results are helpful for the problem that SF-rings are regular rings or not.

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Available abstract

Aim A ring R is a left(right) SF'-ring if every simple singular left(right) R-module is flat.The regularity of SF'-rings is studied.Methods SF'-rings are discussed on condition that every idempotent element is left semi-center and SF'-rings are LANE-ring.Results Two new characterizations of regular rings are given.(1) If R is a left SF'-ring and R/Z(RR) is reduced,then R is strongly regular;(2) R is strongly regular if and only if R is a left SF'-ring whose every idempotent element of R is left semi-center and is LANE-ring.Conclusion The results are helpful for the problem that SF-rings are regular rings or not.

Key concepts: Von Neumann regular ring, Idempotence, Ring (chemistry), Mathematics, Primitive ring, Center (category theory), Element (criminal law), Reduced ring

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