On Weakly Regular Rings and SSF-rings

In this work we consider weakly regular rings whose simple singular right R-Modules are flat. We also consider the condition (*): R satisfies L(a)r(a) for any aR. We prove that if R satisfies(*) and whose simple singular right R-module are flat, then Z (R) the center of R is a von Neunann regular ring. We also show that a ring R either satisfies (*) or a strongly right bounded ring in which every simple singular right R-module is flat, then R is reduced weakly regular rings.