The Regularity Of GP-V-rings And SF-rings

Since Carl Faith introduced the V -ring conception, many algebraic scholars have studied it thoroughly. Following the V - ring, the French mathematician Roger.Yue Chi Ming introduced P - injective conception and GP - injective (i.e.YJ - injective), subsequently, V - rings are naturally extended to P - V- ring and GP - V- ring.In this paper, we mainly discuss the regularity of GP - V- rings and SF - rings. We also study the regularity of GP - V'- rings and SGPF rings which are separately the generalizations of them. There are four chapters on the whole. In the first foreword chapter, we introduce the history of how V - rings develop to GP - V- rings and the main results of this paper. We also introduce some chief conceptions used in this paper. In the second chapter, we mainly discuss the regularity of GP - V- rings and GP - V'-rings. Our chief purpose is to weaken the condition that one-sided ideal is an ideal(We substitute weaker condition which is quasi-ideal or generalized weakly ideal for the ideal) and discuss the biregularity and strong regularity of GP - V'-rings and GP - V- rings; we get that R is strongly regular if and only if R is an abelian GP - V'-ring and every maximal essential left ideal is an quasi-ideal. In the third chapter, we study the regularity of SF - rings. Meanwhile, we extend some results of SF - rings to SGPF rings, and obtain that if R is a left SGPF ring and every maximal left ideal is a generalized weakly ideal or a quasi-ideal, then R / J ( R ) is strongly regular. In the fourth chapter, we firstly recur to idempotent reflexive, generalized weakly ideal and quasi-ideal to discuss the regularity of GP - V'-rings. We obtain that R is idempotent reflexive semi- abelian left GP - V'- rings and every maximal essential left ideal is an quasi-ideal or generalized weakly ideal or weakly right ideal, then R is a reduced left weakly regular. We also gain some good results. At the end of this chapter, we get several theorems on the regularity of n - P - V- rings and n - P - V'- rings.