von-Neumann正则环与左SF-环

On von-Neumann Regular Rings and Left SF-Rings

中文关键词:  正则环  强正则环  SF-环  广义弱理想

英文关键词:regular ring  strongly regular ring  SF-ring  generalized weak ideal.

中文摘要:

      环R称为左SF-环,如果每个单左R-模是平坦的.众所周知,Von-Neumann正则环是SF-环,但SF-环是否是正则环至今仍是公开问题,本文主要研究左SF-环是正则环的条件,证明了:如果R是左SF-环且R的每个极大左(右)理想是广义弱理想,那么R是强正则环.并且推广了Rege[3]中的相应结果.

英文摘要:

      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].