Semi-Regular Bi-Modules

Abstract

In this paper, we study the bi-module of the form , where  are right (left) modules over a some ring , as a generalization of endomorphism of module. Where we study a semi-regular concept of the bi-module  and its relation with direct projective and direct  injective modules.

Where we proved that a many of necessary and sufficient conditions to be the module  semi-regular.

Furthermore, we proved that the bi-module  is semi-regular if and only if  is direct projective module and for every nonzero element ,  contains a nonzero direct summand of .

Also, we proved that the bi-module  is semi-regular if and only if  is direct injective module and for every nonzero element ,  contained in a direct summand  of .

Finally, we proved that the bi-module  is module if and only if  is direct projective module. Also, we proved that the bi-module  is module if and only if  is direct injective module.