On A Generalization Of V-Rings

V-ring is an important conception of ring theory,many mathematic authors all the world have made extensive discussion on V-rings.As generalization, main aim of this paper are to discuss regularities or strongly regularities of GP-V-ring,GP-V -ring and SPF-ring.In the whole, this paper has three chapters. The first chapter, main instead " duo-ring " condition of " every maximal left ideal is GW-ideal " condition,study strongly regularities of GP-V-ring on this condition.lt is shown that (1) R is strongly regular iff R is left GP-V-ring whose maximal left ideals are GW-ideal. (2)R is strongly regular iff R is left GP-V-ring whose maximal right ideals are GW-ideal. The second chapter, generalize some results of GP-V-ring to GP-V-ring, discuss regularity of GP-V'-ring.It is shown that (1) R is left self-injective regular with non-zero socle iff R is left GP-V -ring with Soc(RR) = Soc(Rr) and R contains an injective maximal left ideal. (2)R is regular ring and every maximal essential left ideal is ideal iff R is left GP-injective left GP-V -ring and every maximal essential left ideal is ideal. The third chapter, main study if R is SPF-ring ,the strongly regularity of R/J(R).It is shown that (l)If R is left SPF-ring whose every maximal left ideal is GW-ideal,then R/J is strongly regular. (l)If R is left SPF-ring whose every maximal right ideal is GW-ideal,then R/J is strongly regular.