PI-extending modules via nontrivial complex bundles and Abelian endomorphism rings

A module is said to be PI-extending provided that every projection invariant submodule is essential in a direct summand of the module. In this paper, we focus on direct summands and indecomposable decompositions of PI-extending modules. To this end, we provide several counter examples including the tangent bundles of complex spheres of di- mensions bigger than or equal to 5 and certain hyper surfaces in projective spaces over complex numbers and obtain results when the PI-extending property is inherited by direct summands. Moreover, we show that under some module theoretical conditions PI-extending modules with Abelian endomorphism rings have indecomposable decompositions. Finally, un- der suitable hypotheses, we apply our former results to obtain that the nite exchange property implies the full exchange property.