On categories of projective modules

The authors have studied some structures in categories of completely indecomposable modules in [5], [6] and [7], respectively.Furthermore, one of the authors has given some characterization of semi-perfect modules, defined in [9], in terms of semi-Γ-nilpotent system in [6].In this note, we shall work in the same frame and give generalizations of some results in [6], [9] and [11].Let R be a ring with identity and %Jl R the category of R -right modules.By Sί (resp.Sly) we denote the full sub-additive category of -SHi^, whose objects consist of all R (resp.i?-finitely generated)-projective modules and we denote the Jacobson radical of SI by 3ί or /(SI), (see the definition in [3], [6] and [8]).Then we shall show, in the first section, that SΪ//(St) (resp.31^/(21^)) is a C 3completely reducible (resp.completely reducible artinian) abelian category if and only if R is a right (resp.semi-) perfect ring, defined in [1].In the second section, we shall study a directsum of projective modules P-Σ 0P Λ , and show