The Injectivity Of Modules And Their Relations

This thesis mainly investigated the properties of some classes of special injective modules and their relationships.The first chapter introduced the concepts of injective modules, such as GQP-injective modules, AQP-injective modules, AGP-injective modules, APQ-injective modules,QPQ-injective modules. And we recall some valuable results on them.In chapterâ…¡, we introduced the definitions of quasi GP-injective modules,PQ-injective modules,quasi AP-injective modules,APQ-injective modules,AQP-injective modules,QPQ-injective modules,GQP-injective modules, and discussed their relations. By definitions of QP-injective modules and GQP-injective modules we know that the QP-injective modules are GQP-injective modules, the ring R is right P-injective ring if and only if the module RR is QP-injective module, and the ring R is GP-injective ring if and only if the module RR is GQP-injective module. Since GP-injective rings are not necessarily P-injective ring, GQP-injective module may not be QP-injective module.The two special injective modules, i.e. GQP-injective modules and AQP-injective modules, are dicussed in the third chapter. Meanwhile, we investigated the GQP-injectivity of a module and the PQ-injectivity of it's endomorphism ring, the AQP-injectivity of a module and the AP-injectivity of it's endomorphism ring. Through the study of W(S) and J(S),we know that the GQP-injective modules and AQP-injective modules shared the same property. Furthermore, we realized the M-cyclic submodule paly an important role by investigating the elements of S and W(S).