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Almost  N -projective modules and generalized  N -projective modules play an important role in the study of lifting modules and artinian rings. In this paper, we study a relationship between almost  N -projective modules and generalized  N -projective modules, and give new characterizations of almost  N -projective modules and generalized  N -projective modules by homomorphisms between their projective covers, respectively. Moreover, using the result, we consider a condition for a module which is almost-projective for anyto be almost-projective.

Almost projective modules and generalized projective modules