Ann-Coherent Rings, Fann-Flat Modules And Fann-Injective Modules

In this paper , three points are chiefly discussed.(i) We initially introduce the notion of ann-coherent rings and study theproperties of ann-coherent rings . We then discuss the relation between ann-coherent rings and some special rings , for example , coherent rings,Π- coherentrings,AF rings and so on. In addition , we study ann-coherent rings andtheir signification for the duality theory , and show the reflexitivity of ideals andquotient modules of ann-coherent rings under some conditions.(ii) The concept of flat modules is generalized to fann-flat modules . Welist a series of equivalent characterizations and properties of fann-flat modules .We also extend the famous Chase's theory into ann-coherent rings by fann-flatmodules . From this , we can see the relation between ann-coherent rings andfann-flat modules . At the same time , we use fann-flat modules to charater VNregular rings.(iii) The concept of injective modules is generalized to that of fann-injectivemodules . We also introduce the notion of fann-injective rings and study theequivalent characterizations and properties of fann-injective modules and fann-injective rings . Then we discuss the relation between fann-injective modules andfann-flat modules , which generalizes the results on the Lambek rule . It is fromthe study of these that so much is learnt about fann-injective modules .