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.