$pi$-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 dimensions 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, we apply our former results, getting that, under suitable hypotheses, the nite exchange property implies the full exchange property.