GENERALIZATIONS OF PRINCIPALLY QUASI-INJECTIVE MODULES AND QUASIPRINCIPALLY INJECTIVE MODULES

LetR be a ring andM a rightR-module with S = End(MR). ThemoduleM 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, respec-tively. Specially in the case RR, we obtain some results on AP-injective rings as corollaries. 1