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2004•Journal of Mathematical Research and ExpositionRequires access

On von-Neumann Regular Rings and Left SF-Rings

Xiaodong Wang

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Abstract

A rings R is defined to be a left SF-ring if all simple left R -modules are flat. It is known that von Neumann regular rings are SF-rings. But the question whether a left SF-ring is necessarily regular remains open. In this paper, we study the regularity of left SF-rings which satisfy certain additional conditions. It is proved that R is strongly regular if R is a left SF-ring whose maximal left (right) ideals are generalized weak ideals. Furthermore, we generalize some results in [3].

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A rings R is defined to be a left SF-ring if all simple left R -modules are flat. It is known that von Neumann regular rings are SF-rings. But the question whether a left SF-ring is necessarily regular remains open. In this paper, we study the regularity of left SF-rings which satisfy certain additional conditions. It is proved that R is strongly regular if R is a left SF-ring whose maximal left (right) ideals are generalized weak ideals. Furthermore, we generalize some results in [3].

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Available abstract

A rings R is defined to be a left SF-ring if all simple left R -modules are flat. It is known that von Neumann regular rings are SF-rings. But the question whether a left SF-ring is necessarily regular remains open. In this paper, we study the regularity of left SF-rings which satisfy certain additional conditions. It is proved that R is strongly regular if R is a left SF-ring whose maximal left (right) ideals are generalized weak ideals. Furthermore, we generalize some results in [3].

Key concepts: Von Neumann regular ring, Mathematics, Primitive ring, Ring (chemistry), Von Neumann architecture, Combinatorics, Left and right, Simple ring

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