TY - JOUR ID - 999 TI - $PI$-extending modules via nontrivial complex bundles and Abelian endomorphism rings JO - Bulletin of the Iranian Mathematical Society JA - BIMS LA - en SN - 1017-060X AU - Kara, Y. AU - Tercan, Adnan AU - Yaşar, R. AD - Hacettepe University‎, ‎Faculty of Science‎, ‎Department of Mathematics‎, ‎06532‎, ‎Beytepe‎, ‎Ankara‎, ‎Turkey. AD - Hacettepe University Department of Mathematics Y1 - 2017 PY - 2017 VL - 43 IS - 1 SP - 121 EP - 129 KW - extending module KW - projective invariant KW - tangent bundle KW - exchange property DO - N2 - 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 finite exchange property implies the full exchange property. UR - http://bims.iranjournals.ir/article_999.html L1 - http://bims.iranjournals.ir/article_999_9307a1ce4337ba7bd09adbd5dc888465.pdf ER -