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.
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