Generalizations of principally quasi‐injective modules and quasiprincipally injective modules

Let R be a ring and M a right R‐module with S = End(MR). The module M is called almost principally quasi‐injective (or APQ‐injective for short) if, for any m ∈ M, there exists an S‐submodule Xm of M such that lMrR(m) = Sm ⊕ Xm. The module M is called almost quasiprincipally injective (or AQP‐injective for short) if, for any s ∈ S, there exists a left ideal Xs of S such that lS(Ker(s)) = Ss ⊕ Xs. In this paper, we give some characterizations and properties of the two classes of modules. Some results on principally quasi‐injective modules and quasiprincipally injective modules are extended to these modules, respectively. Specially in the case RR, we obtain some results on AP‐injective rings as corollaries.