On Regularity and Flatness

Nazar Shuker

doi:10.33899/csmj.2004.164107

Abstract

A ring R is called a right SF-ring if all its simple right R-modules are flat. It is well known that Von Neumann regular rings are right and left SF-rings. In this paper we study conditions under which SF-rings are strongly regular. Finally, some new characterstic properties of right SF-rings are given.

Highlights

In this paper all rings are assumed to be associative with identity, and all modules are unital right R-modules
It is well known that a ring R is Von Neumann regular if and only if every right R-module is flat [3]
We recall that: 1- A ring R is called reduced if R contains no non-zero nilpotent elements. 2-R is said to be Von Neumann regular if a aRa for every a R, and R is called strongly regular if a a2R

Summary

INTRODUCTION

In this paper all rings are assumed to be associative with identity, and all modules are unital right R-modules. Following [2], a ring R is called a right (left) SF-ring if all of its simple right (left) R-modules are flat. It is well known that a ring R is Von Neumann regular if and only if every right (left) R-module is flat [3]. Ramamurthi in [8] asked whether left and right SF-ring is Von Neumann regular. 2-R is said to be Von Neumann regular (or just regular) if a aRa for every a R, and R is called strongly regular if a a2R. 5-Following [9], for any ideal I of R, R/I is flat if and only if for each a I , there exists b I such that a=ba. 6-Y and J will stand for the right singular ideal and Jacobson radical of R

RINGS WHOSE SIMPLE MODULES ARE FLAT

BASIC PROPERTIES