Pure Quasi-injective And Pure Quasi-projective Modules

In Homologic Algebra,the quasi-injective modules and quasi-projective modules play impotant roles.The main purpose of this thesis is to extend them and give some further properties of pure quasi-projective modules and pure quasi-injective modules.In the first chapter,the pure injective module is extened to the pure quasiinjective module.Firstly,We prove that the module⊕M_i∈_R M is a pure quasimodule if and only if each M_i is a pure M_j-module,where⊕M_i is a regular module.Furthermore,we investigate the relations between the pure quasi-injective and quasi-injective modules,we get a new properties that M is a pure quasi-injective module if and only if it is a pure injective module,where the module M is a pure generator.Next,we prove that L-rings,v-rings,regular rings and SF-rings coincide with each other under the comutative case.In the second chapter,the pure projective module is extened to pure quasiprojective module.We introduce the pure quatient module and show that the image of a pure epimorphism in a pure factor module is still a pure factor module, and characterise pure quasi-projective modules by means of the pure factor module. Simultaneously,I prove that a submodule U of M is a pure quasi-projective module if and only if(?)f∈End(M),f(u)∈U,and we characterise several useful rings by pure quasi-projective modules.In the last chapter,we introduce the pure quasi-injective resolution and the pure quasi-projective resolution.It is proved that a module M is a pure quasi-injective module if and only if the functor PqExt_R~1(-,M)=0 for all modules.Finally,we obtain some equivalent characterizations of pure quasi-injective dimensions,which are useful in the classifications of rings.