Weakly np- Injective Rings and Weakly C2 Rings

A ring R is called left weakly np injective if for each non-nilpotent element a of R, there exists a positive integer n such that any left R-homomorphism from Ra-n to R is right multiplication by an element of R. In this paper various properties of these rings are first developed, many extending known results such as every left or right module over a left weakly np injective ring is divisible; R is left seft-injective if and only if R is left weakly np-injective and R-R is weakly injective; R is strongly regular if and only if R is abelian left pp and left weakly np injective. We next introduce the concepts of left weakly pp rings and left weakly C2 rings. In terms of these rings, we give some characterizations of (von Neumann) regular rings such as R is regular if and only if R is n regular, left weakly pp and left weakly C2. Finally, the relations among left C2 rings, left weakly C2 rings and left GC2 rings are given.