| As we all know, projective module and injective module are the basic modules in rings theory and modules theory, and also are the main objects homological algebra and modules theory's study. They are applied in algebraic geometry's study. So some generalizations on the two modules are studied and discussed in the three chapters ofthis article.In the first chapter, some basic concepts of injective modules are introduced firstly, then the concept of essentially GP—injective module introduced, and some properties of essentially GP—injective module are discussed, then a series of theorems are given. The same time, the essentially GP-injective decomposition of a left R-module is given, and also essentially GP-injective dimension and essentially GP-injective integrated dimension are defined.In the second chapter, some basic concepts of projective modules are introduced, then the concept of essentially M—projective modules are given, secondly, some equal conditions of essentially M—projective modules are given, some properties of essentially M—projective modules are discussed. Moreover, the essentially M—projective decomposition of a left R—module is given, and also essentially M—projective dimension and essentially M—projective integrated dimension are defined. At last, a new ring, named E—semisimple ring, is constructed, and E—semisimple ring is characterized by essentially M—projective module and essentially M—projective dimension.In the third chapter, some studies on quasi-projective modules and quasi-injective modules are done, and quasi-injective modules are the generalization of injective modules, quasi-projective modules are the generalization of projective modules, The properties of quasi-injective modules and quasi-prcjective modules have been studied by many scholars. So in this chapter, the main results are the generalization of Schanuel lemma on quasi-injective modules and quasi-projective modules. At last, some properties of semisimple modules are gained.
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