On Von Neumann Regular Rings And SF-Rings

A ring R is said to be a left(right)S F-ring if each simple left(right)R-module is flat.As we know,von Neumann regular rings are both left and right S F-rings.In this thesis,we study conditions under which S F-rings are von Neumann regular and characterizations and properties of von Neumann regular rings and S F-rings.In Chapter 2,we investigate the functions of GW-ideals,W-ideals and weakly left(right)ideals in the matter of the study of the regularity of S F-rings.We prove that concepts of these weakened ideals are different in generalizing concept of ideals,however,they perform the same functions in the study of regularity of S F-rings.In Chapter 3,new characterizations of von Neumann regular rings and S F-rings are given.The following statements are shown to be equivalent for a ring R:(1)R is von Neumann regular;(2)Every principal left ideal of R is a projective left annihilator of an element of R;(3)R is a left p.p.right S F-rings.In addition,for a ring R and a ∈R,we introduce a multiplication abelian group Ta{b|a=ab,b ∈ R)which plays a critical role in the characterization of both von Neumann regular rings and S F-rings.In the last chapter,we discuss some important properties of von Neumann regular rings and S F-rings.