$PI$-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.

Keywords

Main Subjects


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Volume 43, Issue 1
January and February 2017
Pages 121-129
  • Receive Date: 05 November 2014
  • Revise Date: 20 October 2015
  • Accept Date: 20 October 2015