Abstract: Let R be an associative（not necessary commutative） ring. In this paper,we give some properties of Hom-faithful and Hom-cofaithful R-modules. We investigate Σ-self-orthogonal R-modules and P_C-projective dimensions, and give a characterization when the inequality P_C-pd V max{P_C-pd U, P_C-pd W } is strict for an exact sequence 0 → U → V → W → 0. Dually, we study Π-self-orthogonal modules and I_C-injective dimensions, and give a similar characterization when the inequality I_C-id V max{I_C-idU, I_C-idW } is strict. 

Key words: Hom-faithful, Hom-cofaithful;  Σ-self-orthogonal; Π-self-orthogonal