Modules Close to the Quasi-projective

In this paper, we study modules that are natural extensions of the class of quasi-projective modules. In particular, the paper studies small quasi-projective modules and classes of modules related to them. It is shown that over right perfect rings, the class of small quasi-projective right modules coincides with several classes of right modules close to projective modules studied in the article. Small quasi-projective abelian groups and related classes of abelian groups are also studied.