On Von Neumann Regular Rings And IF Rings

Let R be a ring,R is said to be a left(right)IF ring if every injective left(right)R-module is flat.R is said to be a von Neumann regular ring if for every a E R,there is a’∈R such that a=aa’a.The structure of this thesis is as follows:The first part introduces some preliminaries and known results.The second part mainly discusses the relation between the injectivity of Hom(_,_)and von Neumann regular rings.For a commutative ring R,it is shown that R is a von N eumann regular ring iff HomR(F,C)is injective for all flat R-modules F and all cotorsion R-modules C.The last part mainly discusses the relation between the flatness of Hom(_,_)and IF rings.For a commutative IF ring R,it is proven that R is a von Neumann regular ring iff HomR(F,M)is flat for all injective R-modules F and all R-modules M.