Study About The Generalized Injective Modules

As everyone knows, injective modules are played a more and more important role in the module. In the injective module, the module M is a CS-module if every submodule is essential in a direct summand. This thesis does some generalization of CS-module, introduced the P-CS module and P-uniform module. This thesis is made up of four chapters:In chapter one, we give some prepare knowledge, motivations and main results.In chapter two, we defined a P-CS module and P-CS submodule, and given a new characterization of the relationship between them, proved the equal condition of P-CS module to keep direct and direct-summand properties. Next, We defined a P-continuous module and quasi-P-continuous module. We proved the hereditary of P-continuous module and quasi-P-continuous module.In chapter three, we introduced P-uniform module, proved that some of the links between the P-uniform module with their essentially cyclic submodules and P-uniform submodule. Further we defined the P-uniform dimension and given the existence of P-uniform dimension. Then the P-uniform dimension is used to characterize some related properties of the P-uniform module.In chapter four, we introduced the connection in P-CS module and P-uniform mod-ule, then we proved the sufficient condition that a finite direct sum of P-uniform mod-ules is P-CS module.