On generalization of quasi-principally injective modules

A module [Formula: see text] is said to be generalized quasi-principally injective (briefly, GQP-injective) (respectively, generalized almost quasi-principally injective (briefly, generalized AQP-injective)) if, for any [Formula: see text], there exists a positive integer [Formula: see text] (respectively, there exist a positive integer [Formula: see text] and a left ideal [Formula: see text]) such that [Formula: see text] and [Formula: see text] (respectively, [Formula: see text]). In this paper, we give some characterizations and properties of the classes of generalized AQP-injective modules. Some results on quasi-principally injective modules are extended to these modules. Some properties of GQP-injective modules are studied.