Mohammad Ashraf, Aisha Jabeen, Mohammad Afajal Ansari, Mohd Shuaib Akhtar

DOI Number
First page
Last page


Let $\mathrm{R}$ be a commutative ring with unity, $\mathrm{A},\mathrm{B}$ be $\mathrm{R}$-algebras and $\mathrm{M}$ be an $(\mathrm{A}, \mathrm{B})$-bimodule. Let $\mathfrak{T}=Tri(\mathrm{A},\mathrm{M},\mathrm{B})$ be a $(n-1)$-torsion free triangular algebra. In this article, we prove that every multiplicative Lie $n$-higher derivation on triangular algebras has the standard form. Also, the main result is applied to some classical examples of triangular algebras such as upper triangular matrix algebras and nest algebras.


Triangular algebras, Lie type derivation.

Full Text:



bibitem{Ab92} {sc I. Z. Abdullaev}: textit{$n$-Lie derivations on von Neumann algebra}. Uzbek. Mat. Zh. {bf5} (1992), no. 6, 3-–9.

bibitem{AJ15} {sc M. Ashraf {rm and} A. Jabeen}: textit{Nonlinear Jordan triple derivable mappings on triangular algebras}. Pac. J. Appl. Math. {bf7} (2015), no. 4, 225–-235.

bibitem{AJ3} {sc M. Ashraf {rm and} A. Jabeen}: Contemp. math., Advances in rings and modules, vol. 715, ch. textit{Nonlinear Lie triple higher derivation on triangular algebras}. pp. 43–-57, Amer. Math. Soc., Providence, RI, 2018.

bibitem{AJSA} {sc M. Ashraf {rm and} A. Jabeen}: textit{Nonlinear Jordan triple higher derivable mappings of triangular algebras}. Southeast Asian Bull. Math. {bf42} (2018), no. 1, 503-–520.

bibitem{AWW19} {sc M. Ashraf, B. A. Wani {rm and} F. Wei}: textit{Multiplicative $ast$-Lie triple higher derivations of standard operator algebras}. Quaest. Math. {bf42} (2019), no. 7, 857-–884.

bibitem{BE12} {sc D. Benkoviv{c} {rm and} D. Eremita}: textit{Multiplicative Lie $n$-derivations of triangular rings}. Linear Algebra Appl. {bf436} (2012), no. 11, 4223–-4240.

bibitem{Ch00} {sc W. S. Cheung}: textit{Maps on triangular algebras}. Ph.D. dissertation, University of Victoria, 2000.

bibitem{Da91} {sc M. Daif}: textit{ When is a multiplicative derivation additive?}. Internat. J. Math. Sci. {bf 14} (1991), 615–-618.

bibitem{DL17} {sc Y. Ding {rm and} J. Li}: textit{Lie $n$-higher derivations and Lie $n$-higher derivable mappings}. Bull. Aust. Math. Soc. {bf96} (2017), no. 2, 223-–232.

bibitem{FWX13} {sc A. Fov{s}ner, F. Wei {rm and} Z. Xiao}: textit{Nonlinear Lie-type derivations of von Neumann algebras and related topics}. Colloq. Math. {bf132} (2013), no. 1, 53–-71.

bibitem{Ha14} {sc D. Han}: textit{Lie type higher derivations on operator algebras}. Bull. Iranian Math. Soc. {bf40} (2014), no. 5, 1169–-1194.

bibitem{JA20} {sc A. Jabeen {rm and} M. Ashraf}: textit{Nonlinear $ast$-Lie derivations on unital algebras}. Beitr. Algebra Geom. {bf61} (2020), no. 4, 731–-746.

bibitem{JSM15} {sc A. R. Janfada, H. Saidi {rm and} M. Mirzavaziri}: textit{Characterization of Lie higher derivations on $C^ast$-algebras}. Bull. Iranian Math. Soc. {bf41} (2015), no. 4, 901–-906.

bibitem{JLZ12} {sc P. Ji, R. Liu {rm and} Y. Zhao}: textit{Nonlinear Lie triple derivations of triangular algebras}. Linear Multilinear Algebra {bf60} (2012), no. 10, 1155-–1164.

bibitem{JQ11} {sc P. Ji {rm and} W. Qi}: textit{Characterizations of Lie derivations of triangular algebras}. Linear Algebra Appl. {bf435} (2011), 1137–-1146.

bibitem{LS12} {sc J. Li {rm and} Q. Shen}: textit{Characterzations of Lie higher and Lie triple derivations of triangular algebra}. J. Korean Math. Soc. {bf49} (2012), no. 2, 419-–433.

bibitem{Ma64} {sc W. S. Martindale III}: textit{Lie derivations of primitive rings}. Michigan Math. J. {bf11 } (1964), 183–-187.

bibitem{Ma69} {sc W. S. Martindale III}: textit{When are multiplicative mappings additive?}. Proc. Amer. Math. {bf21} (1969), 695–-698.

bibitem{Mi78} {sc C. R. Miers}: textit{Lie triple derivations of von Neumann algebras}. Proc. Amer. Math. Soc. {bf71} (1978), no. 1, 57–-61.

bibitem{QH10} {sc X. Qi {rm and} J. Hou}: textit{Lie higher derivations on nest algebras}. Commun. Math. Res. {bf26} (2010), no. 2, 131-–143.

bibitem{SR97} {sc R. P. Stanley {rm and} G. C. Rota}: textit{Enumerative combinatorics}. Cambridge Studies in Advanced Mathematics, vol. 1, Cambridge University Press, 1997.

bibitem{Wa37} {sc M. Ward}: textit{Arithmetic functions on rings}. Annals of Mathematics {bf38} (1937), no. 3, 725–732.

bibitem{WX11} {sc F. Wei {rm and} Z. Xiao}: textit{Higher derivations of triangular algebras and its generalizations}. Linear Algebra Appl. {bf435} (2011), no. 5, 1034–-1054.

bibitem{XW12} {sc Z. Xiao {rm and} F. Wei}: textit{Nonlinear Lie higher derivations on triangular algebras}. Linear Multilinear Algebra {bf60} (2012), no. 8, 979-–994.

bibitem{YZ10} {sc W. Yu {rm and} J. Zhang}: textit{Nonlinear Lie derivations of triangular algebras}. Linear Algebra Appl. {bf432} (2010), no. 11, 2953–-2960.

DOI: https://doi.org/10.22190/FUMI210405073A


  • There are currently no refbacks.

© University of Niš | Created on November, 2013
ISSN 0352-9665 (Print)
ISSN 2406-047X (Online)