LetRbe a ring andMa rightR-module withS=End(MR). The moduleMis called almost principallyquasi-injective (orAPQ-injective for short) if, for anym∈M, there exists anS-submoduleXmofMsuch thatlMrR(m)=Sm⊕Xm. The moduleMis called almostquasiprincipally injective (orAQP-injective for short) if, foranys∈S, there exists a left idealXsofSsuch thatlS(Ker(s))=Ss⊕Xs. In this paper, we give somecharacterizations and properties of the two classes of modules.Some results on principally quasi-injective modules andquasiprincipally injective modules are extended to these modules,respectively. Specially in the caseRR, we obtain some resultsonAP-injective rings as corollaries.
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