On Projection Invariant Semisimple Modules

In this paper, we introduce and investigate the notion of projection invariant semisimple modules. Some structural properties of aforementioned class of modules are studied. We obtain indecomposable decompositions of former class of modules under some module theoretical conditions. Moreover, we explore when the finite exchange property implies full exchange property for the class of projection invariant semisimple modules. Finally, we obtain that the endomorphism ring of a projection invariant semisimple modules is a π- Baer ring.