Φ-Injective Modules,Φ-Flat Modules And Application

Injective modules and flat modules play an important role in rings and cat-egories. In [15]. the author gave definitions of Φ-injective modules and0-flat modules, investigating some basic properties of them. In this paper, we will continue to study more basic properties of them, including the relation between0-injective modules and Φ-flat modules, characterizing Φ-coherent rings by0-flat modules.In the first chapter, we characterize a new type of ring called Φ-hereditary ring by Φ-injective modules, obtaining the equivalent conditions of Φ-hereditary rings. Furthermore, similar to the injective radical of modules, we define the Φ-injective radical of modules, discussing the Φ-injective radical of modules on some special rings.In the second chapter, we firstly recall the definition of Φ-flat module. Then we prove that the definition of Φ-flat module according to nonnil ideals is identical to that in [15]. We give a series of equivalent conditions of Φ-flat modules, and reveal the relation between Φ-injective modules and Φ-flat modules, which is the generalized Lambek criterion. In addition, we obtain the equivalent conditions and basic properties of Φ-coherent rings.In the last chapter, we firstly introduce Φ-injcetive dimension and Φ-flat dimension. Drawing from that injective dimension and flat dimension can be characterized by Ext and Tor, so a new type of ring called NQ ring is introduced. Furthermore, we prove that Φ-injective dimension and Φ-flat dimension can be characterized by Ext and Tor if R is a NQ ring. Then some properties and results of Φ-injective dimension and Φ-flat dimension are given.