Projective Properties Of Module And K Groups

Projective module and injective module play very important roles in ho-mological theory. In this thesis, we discuss the projective properties of modules and their Grothendieck groups.In the first chapter, we study the properities of quasi-principally projective modules. The main results are proved as follows:Theorem 1.14 Let M be a quasi-principally projective module and s,t ∈S = End(M_R).(a) If s(M) can be embeded into t(M), then sS can be embedded into tS.(b) If t(M) is an image of s(M), then tS is an image of sS.(c) If s(M) (?) t(M), then tS (?) sS.Let M be a right R — module , s G S = End(M_R). Let △_s = {t ∈ S|Im(t) (?)Im(s)},Theorem 1.16 Let M be a right R-module and S = End(M_R). Then the following conditions are equivalent:(a) M is quasi-principally projective.(b) △_s = sS for all s G S.(c) If Im(t) (?)Im(s), then tS C sS for all s,t ∈ S.In the second chapter, the author introduce pure projective module, and discuss the relations between pure projective module and pure stably free module. We obtain a necessary and sufficient condition under which a finitely generated pure projective module is pure stably free.Theorem 2.10 Let P be a finitely generated pure projective R—module.Then the following are equivalent:(a) P is pure stably free module .(b) P is the kernel of a surjective R-morphism σ : M → N,M,N are finitely generated pure free modules.