On np-injective rings and Modules - Ahmed , Ferman A.; Abdul-Jabbar, Abdullah M. (International Journal of Membrane Science and Technology, 2023)
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 .