Relative Faithful Exact Functors and Their Applications to Homological Modules

The notions of faithfully projective, faithfully flat, and faithfully injective modules–defined as modules for which the three classical homological functors are both faithful and exact–play fundamental roles across various areas of algebra. In this paper, we extend these notions to the setting of $w$-operation theory. By introducing the concept of $w$-faithfully exact functors, we define and investigate the notions of $w$-faithfully projective, $w$-faithfully flat, and $w$-faithfully injective modules. We establish their fundamental properties and demonstrate their effectiveness in generalizing classical results.