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Left Ideal Preserving Maps on Triangular Algebras

  • Hoger GhahramaniEmail author
Research Paper
  • 1 Downloads
Part of the following topical collections:
  1. Mathematics

Abstract

Let \({\mathcal {A}}\), \({\mathcal {B}}\) be unital algebras, \({\mathcal {M}}\) be an \(({\mathcal {A}},{\mathcal {B}})\)-bimodule and \({\mathcal {T}}=\begin{pmatrix} {\mathcal {A}}&{} {\mathcal {M}}\\ 0 &{} {\mathcal {B}}\end{pmatrix}\) be the corresponding unital triangular algebra over a commutative unital ring \({\mathcal {R}}\). In this paper, we study whether every \({\mathcal {R}}\)-linear map on \({\mathcal {T}}\) that leaves invariant every left ideal of \({\mathcal {T}}\) is a left multiplier, and give some necessary or sufficient conditions for a triangular algebra to have this property. We also give various examples illustrating limitations on extending some of the theory developed. We then apply our established results to generalized triangular matrix algebras and block upper triangular matrix algebras. Moreover, we introduce some algebras other than triangular algebras on which every \({\mathcal {R}}\)-linear map is a left multiplier.

Keywords

Left multiplier Local left multiplier Left ideal preserving Triangular algebra Generalized triangular matrix algebras Block upper triangular matrix algebras 

Mathematics Subject Classification

15A86 16S50 16D99 16S99 

Notes

Acknowledgements

The author thanks the referee and Professor M. N. Ghosseiri for careful reading of the manuscript and for helpful suggestions.

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Copyright information

© Shiraz University 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KurdistanSanandajIran

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