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.
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The author thanks the referee and Professor M. N. Ghosseiri for careful reading of the manuscript and for helpful suggestions.
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Ghahramani, H. Left Ideal Preserving Maps on Triangular Algebras. Iran J Sci Technol Trans Sci 44, 109–118 (2020). https://doi.org/10.1007/s40995-019-00794-2
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DOI: https://doi.org/10.1007/s40995-019-00794-2
Keywords
- Left multiplier
- Local left multiplier
- Left ideal preserving
- Triangular algebra
- Generalized triangular matrix algebras
- Block upper triangular matrix algebras