GENERALIZATION OF VON-NEUMANN REGULAR RINGS TO VON-NEUMANN REGULAR MODULES

Abstract

An element r in a commutative ring R is called regular if there exist s∈R such that rsr=r. Ring R is called vN (von-Neumann)-regular ring if every element is regular. Recall that for any ring R always can be considered as module over itself. Using the fact, it is natural to generalize the definition of vN-regular ring to vN-regular module. Depend on the ways in generalizing there will be some different version in defining the vN-regular module. The first who defined the concept of regular module is Fieldhouse. Secondly Ramamuthi and Rangaswamy defined the concept of strongly regular module of Fieldhouse by giving stronger requirement. Afterward Jayaram and Tekir defined the concept of vN-regular module by generalizing the regular element in ring to regular element in R-module M. In this paper we investigate the properties of each module regular and the linkages between each vN-regular module.