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

Document Type : Research Paper


1 Hacettepe University‎, ‎Faculty of Science‎, ‎Department of Mathematics‎, ‎06532‎, ‎Beytepe‎, ‎Ankara‎, ‎Turkey.

2 Hacettepe University Department of Mathematics


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.


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