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

Document Type : Research Paper

Authors

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

2 Hacettepe University Department of Mathematics

Abstract

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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References

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Bulletin of the Iranian Mathematical Society
Volume 43, Issue 1
February 2017
Pages 121-129

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History

  • Receive Date: 05 November 2014
  • Revise Date: 20 October 2015
  • Accept Date: 20 October 2015

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