The Regularity Of SF-rings

The regularity of GP-V′rings and SF-rings are studied. Some new characterizations of strongly regular rings are given. We get the following results: (1) A ring R is strongly regular if and only if R is a left GP-V′ring whose every maximal essential left ideal is generalized weakly ideal and every idempotent element is left semi-center; (2) If every maximal essential left ideal of R is generalized weakly ideal, then R is regular if and only if R is a left GP-V′, left generalized semi-regular ring with J ( R)(?) Z(RR); (3) A ring R is strongly regular if and only if R is a left GP-V′ring whose every maximal essential left ideal is weakly right ideal and every idempotent element is left semi-center; (4) A ring R is strongly regular if and only if R is a left SF-ring whose every maximal essential left ideal is generalized weak ideal; (5) A ring R is strongly regular if and only if R is a left SF-ring whose l (a) is generalized weak ideal for any a∈{x|x2 =0,x∈R}; (6) R is a left self-injective strongly regular ring with S≠0 if and only if R is a right SF-ring containing an injective maximal left ideal and S (?) Soc( RR); (7) A ring R is strongly regular if and only if R is a left SF-ring and N-ring; (8) A ring R is strongly regular if and only if R is a left SGPF′-ring whose every maximal essential left ideal is weakly right ideal and any idempotent element of R is left semi-center; (9) If R is a left SGPF′-ring whose every maximal left ideal is generalized weakly right ideal, then R /J is strongly regular.