Relactive Covers And Envelopes

This thesis is divided into three parts, in the first part, we define P-projective (pre)-envelopes more generally then projective (pre)-envelopes and investigate the cokernel of P-projective (pre)envelope, i.e., PP-projective modules, and get some characterizations of PP-projective modules. We observe that PP-projective modules are more closer to projective modules than P-projective modules. If every R-module has monic P-projective (pre)envelope, we prove that R is a QF ring if and only if every PP-projective module is projective module. Moreover, we get some characterizations under which every PP-projective module is a projective module as follows:Theorem 1.2.3 Let B → C → 0 be a left R-module exact sequence. Then the following conditions are equivalent:(1) For every PP-projective left R-module M,Hom(M, B) → Hom{M, C) → 0 is exact.(2) There exists epicmorphism σ : K → P₀(A),such that g = εAσ.(3) g can factor through a P-projective left R-module P, i.e., there exists a P-projective left R-module P and homomorphisms f : K → P,h: P → A such that g = hf.(4) There exists L (?) Kerg,such that K/L is P-projective module. Finally, we introduce P-projective dimension of modules and investigate the conditions under which every PP-projective module is projective module. P-projective dimension of a module is also discussed.