Recently, Zhang and Qi introduced and studied the concept of uniformly S-projective (u-S-projective) modules, where S represents a multiplicative subset of a ring. In this paper, we first derive a u-S version of the well-known result that a module is projective if and only if it is a direct summand of a free module. We then provide a characterization of u-S-projective modules in terms of projective modules, specifically when S is regular, and then extend the projective basis lemma to the u-S setting. Finally, we show that a u-S-projective u-S-cover is characterized by a u-S-superfluous submodule, analogous to the way a projective cover is characterized by a superfluous submodule.

Keywords: u-S-exact sequence, u-S-isomorphism, u-S-projective module