On np-injective rings and Modules

Abstract

A given R-module  is called a right np-injective module if for any non-nilpotent element  of , and any right R-homomorphism  can be extended to , if  is np-injective, then  is a right np-injective ring. A given ring  is called right weakly np-injective if for each non-nilpotent element  of , there exists a positive integer  such that any right R-homomorphism  can be extended to . A given ring  is a right weakly np-injective ring, if . In the matrix ring, we poved that  is right np-injective, for some , if  is right Weakly np-injective. So, We extended many of the known properties and characterizations of right np-injective rings and modules. Finally, the main result in np-injective module found that the R-module  is np-injective module if and only if for any  the short exact sequence  of R-modules,  is also a short exact sequence, where  and .