On SF-rings and Regular Rings

. A ring R is called a left (right) SF-ring if simple left (right) R-modules areat. It is still unknown whether a left (right) SF-ring is von Neumann regular. In thispaper, we give some conditions for a left (right) SF-ring to be (a) von Neumann regular;(b) strongly regular; (c) division ring. It is proved that: (1) a right SF-ring R is regularif maximal essential right (left) ideals of R are weakly left (right) ideals of R (this resultgives an armative answer to the question raised by Zhang in 1994); (2) a left SF-ring Ris strongly regular if every non-zero left (right) ideal of R contains a non-zero left (right)ideal of R which is a W-ideal; (3) if R is a left SF-ring such that l(x) (r(x)) is an essentialleft (right) ideal for every right (left) zero divisor x of R, then R is a division ring. 1. IntroductionThroughout this paper, R denotes an associative ring with identity and all ourmodules are unitary. The symbols J(R), Z( R R)(Z(R R )), soc( R R)(soc(R R )) re-spectively stand for the Jacobson radical, left (right) singular ideal and left (right)socle of R. R is semiprimitive if J(R) = 0. R is left non-singular if Z(