On characterizations and properties of nil-injective rings and modules

A special type of injective module which is introduced by Ming is called p-injective if for every a ∈ R, every R-linear map from aR to M can be extended to a linear map from R to M. Then, R is called right p-injective ring if R_R is p-injective. Furthermore, A right R-module M is called nil-injective if for any a ∈ N(R), and any right R-homomorphism f: aR → M can be extended to R → M. Or equivalently, there exists m ∈ M such that f(x) = mx, for all x ∈ aR. If R_R is nil-injective, then R is a right nil-injective ring. In addition, every right p-injective rings are right nil-injective. In the present work, we develop various properties and characterizations of right nil-injective rings and modules, by which many of the known results are extended. Finally, we show that for any R-module M, the R-module M is nil-injective module if and only if for any a∈N(R) the short exact sequence $0→aR→ωR→γR/aR→0$ of R-modules. We can obtain, $0→HomR(R/aR,M)→ΩHomR(R,M)→ΓHomR(aR,M)→0$ is also a short exact sequence, where Γ(f) = fγ and Ω(f) = fω.