Iranian Mathematical Society (IMS)Bulletin of the Iranian Mathematical Society1017-060X43120170222$PI$-extending modules via nontrivial complex bundles and Abelian endomorphism rings121129999ENY.KaraHacettepe University, Faculty of Science, Department of Mathematics, 06532, Beytepe, Ankara, Turkey.AdnanTercanHacettepe University Department of MathematicsR.YaşarHacettepe University, Faculty of Science, Department of Mathematics, 06532, Beytepe, Ankara, Turkey.Journal Article20141105A 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 finite exchange property implies the full exchange property.http://bims.iranjournals.ir/article_999_9307a1ce4337ba7bd09adbd5dc888465.pdf