An R-module m is called strongly D2 if M-(n) is a D2 (equivalently, direct projective) module for every positive integer n. In this paper, we consider the class of quasi-projective R-modules, the class of strongly D2 R-modules and the class of D2 R-modules. We first show that these classes are distinct, which gives a negative answer to the question raised by Li-Chen-Kourki. We also give structural characterizations of strongly D2 modules for finitely generated modules over a principal ideal domain. In addition, we characterize some rings such as Artinian semisimple rings, hereditary rings, semihereditary rings and perfect rings in terms of strongly D2 modules.
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