The Regularity Of Some Special Rings

In this paper, we introduce the concepts of quasi ZI-rings, left(right)WAP-injective rings and generalized N-semiregular rings, and discussregularity of some special rings by means of them. Some properties ofthem and their relationships with some special rings are discussed.In the first chapter, we mainly introduce the concept of quasi ZI-rings, and study the property of quasi ZI-rings and the nonsigularityand regularity of GP-V -rings. We obtain some conclusions as follows:(1)if R is a quasi ZI, left GP-V -ring, then R is left and right non-singular; (2)for a ring R, the following statements are equivalent:(a) R is a strongly regular ring;(b) R is a quasi ZI, MELT, left GP-V -ring;(c) R is a quasi ZI, MERT, right GP-V -ring;(d) R is a quasi ZI, MELT, left GP-V -ring;(e) R is a quasi ZI, MERT, right GP-V -ring.In the second chapter, we mainly give the concept of left(right)WAP-injective rings, and discuss that the strong regularity of the ringwhose the idempotent elements are left semicentral. For a ring R whoseidempotent elements are left semicentral, it proved the following condi-tions are equivalent:(1) R is strongly regular ring;(2) R is a left WAP-injective, left PP-ring;(3) R is a left WAP- injective ring satisfing (?) condition. In the last chapter, we mainly give the concept of generalized N-semiregular rings, and discuss the regularity of generalized N-semiregularrings. It is proved that R is a reduced and generalized N-semiregularring if and only if R is a strongly regular ring.