On CE-injective Module And CE-flat Module

As we all know, injective module and flat module play important roles in module theory and ring theory, and also are the important objects in modules theory and homological theory’ s study. Them also are many applied in algebraic geometry. We do some generalizations on the injective module and flat module in this thesis, and also are discussed and studied in the three chapters.In the first chapter, we introduce the notion of CE-injective modules and CE-flat modules,At the same time,we discussed some properties of CE-injective modules and CE-flat modules. Then a series of important theorems are given. For we investigated the relation of CE-injective modules and CE-flat modules, and proved that IIM, is CE-injective modules if only if so is each M,In the second chapter, we introduce the notion of CE-injective decomposition, andCE-flat decomposition. Further, we defined the CE-injective dimension and CE-flat dimension. Because of injective dimension and flat dimension is always available Ext and Tor engraved, in this chapter we proved that the CE-injective dimension and CE-flat dimension is portrayed using Ext and Tor, so we define the CO-ring, and made some equivalent characterizations of CQ-ring.In the third chapter, we introduce the concept of GFP-injective modules and G-flat modules, then we proved GFP-injective modules and CE-injective modules are equivalent when ring R is CQ-ring, and use them to portray a coherent ring, semi-hereditary ring and G-regular ring.