# Article

 Title: Generalized Higher Derivations on Lie Ideals of Triangular Algebras (English) Author: Ashraf, Mohammad Author: Parveen, Nazia Author: Wani, Bilal Ahmad Language: English Journal: Communications in Mathematics ISSN: 1804-1388 Volume: 25 Issue: 1 Year: 2017 Pages: 35-53 Summary lang: English . Category: math . Summary: Let $\mathfrak {A} = \begin {pmatrix}\mathcal {A} & \mathcal {M}\\ &\mathcal {B} \end {pmatrix}$ be the triangular algebra consisting of unital algebras $\mathcal {A}$ and $\mathcal {B}$ over a commutative ring $R$ with identity $1$ and $\mathcal {M}$ be a unital $\mathcal {(A, B)}$-bimodule. An additive subgroup $\mathfrak { L }$ of $\mathfrak { A }$ is said to be a Lie ideal of $\mathfrak {A}$ if $[\mathfrak {L},\mathfrak {A}]\subseteq \mathfrak {L}$. A non-central square closed Lie ideal $\mathfrak { L }$ of $\mathfrak { A }$ is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on $\mathfrak {A}$, every generalized Jordan triple higher derivation of $\mathfrak {L}$ into $\mathfrak {A}$ is a generalized higher derivation of $\mathfrak {L}$ into $\mathfrak { A }$. (English) Keyword: Admissible Lie Ideals Keyword: triangular algebra Keyword: generalized higher derivation Keyword: generalized Jordan higher derivation Keyword: generalized Jordan triple higher derivation MSC: 15A78 MSC: 16W25 MSC: 47L35 idZBL: Zbl 1390.16039 idMR: MR3667075 . Date available: 2017-09-01T12:14:53Z Last updated: 2020-01-05 Stable URL: http://hdl.handle.net/10338.dmlcz/146843 . Reference: [1] Ashraf, M., Khan, A., Haetinger, C.: On $(\sigma ,\tau )$-higher derivations in prime rings.Int. Electron. J. Math., 8, 1, 2010, 65-79, Zbl 1253.16039, MR 2660541 Reference: [2] Ashraf, M., Khan, A.: On generalized $(\sigma ,\tau )$-higher derivations in prime rings.SpringerPlus, 38, 2012, MR 3166542 Reference: [3] Awtar, R.: Lie ideals and Jordan derivations of prime rings.Proc. Amer. Math. Soc., 90, 1, 1984, 9-14, Zbl 0528.16020, MR 0722405, 10.1090/S0002-9939-1984-0722405-2 Reference: [4] Bergen, J., Herstein, I. N., Kerr, J. W.: Lie ideals and derivations of prime rings.J. Algebra, 71, 1981, 259-267, Zbl 0463.16023, MR 0627439, 10.1016/0021-8693(81)90120-4 Reference: [5] Brešar, M.: On the distance of the composition of two derivations to the generalized derivations.Glasgow Math. J., 33, 1991, 89-93, Zbl 0731.47037, MR 1089958, 10.1017/S0017089500008077 Reference: [6] Chase, S. U.: A generalization of the ring of triangular matrices.Nagoya Math. J., 18, 1961, 13-25, Zbl 0113.02901, MR 0123594, 10.1017/S0027763000002208 Reference: [7] Cortes, W., Haetinger, C.: On Jordan generalized higher derivations in rings.Turkish J. Math., 29, 1, 2005, 1-10, Zbl 1069.16039, MR 2118947 Reference: [8] Ferrero, M., Haetinger, C.: Higher derivations and a theorem by Herstein.Quaest. Math., 25, 2, 2002, 249-257, Zbl 1009.16036, MR 1916335, 10.2989/16073600209486012 Reference: [9] Ferrero, M., Haetinger, C.: Higher derivations of semiprime rings.Comm. Algebra, 30, 5, 2002, 2321-2333, Zbl 1010.16028, MR 1904640, 10.1081/AGB-120003471 Reference: [10] Haetinger, C.: Higher derivation on Lie ideals.Tend. Mat. Apl. Comput., 3, 1, 2002, 141-145, MR 2001254, 10.5540/tema.2002.03.01.0141 Reference: [11] Haetinger, C., Ashraf, M., Ali, S.: On Higher derivations: a survey.Int. J. Math. Game Theory Algebra, 19, 5/6, 2011, 359-379, Zbl 1234.16030, MR 2814896 Reference: [12] Han, D.: Higher derivations on Lie ideals of triangular algebras.Sib. Math. J., 53, 6, 2012, 1029-1036, Zbl 1261.16043, MR 3074440, 10.1134/S0037446612060079 Reference: [13] Hasse, F., Schmidt, F. K.: Noch eine Begründung der Theorie der höheren DiKerentialquotienten einem algebraischen Funktionenköroer einer Unbestimmten.J. reine angew. Math., 177, 1937, 215-237, MR 1581570 Reference: [14] Jing, W., Lu, S.: Generalized Jordan derivations on prime rings and standard operator algebras.Taiwanese J. Math., 7, 4, 2003, 605-613, Zbl 1058.16031, MR 2017914, 10.11650/twjm/1500407580 Reference: [15] Jung, Y. S.: Generalized Jordan triple higher derivations on prime rings.Indian J. Pure Appl. Math., 36, 9, 2005, 513-524, Zbl 1094.16023, MR 2210246 Reference: [16] Lanski, C., Montgomery, S.: Lie structure of prime rings of characteristic 2.Pacific J. Math., 42, 1972, 117-136, MR 0323839, 10.2140/pjm.1972.42.117 Reference: [17] Nakajima, A.: On generalized higher derivations.Turk. J. Math., 24, 3, 2000, 295-311, Zbl 0979.16022, MR 1797528 Reference: [18] Xiao, Z. H., Wei, F.: Jordan higher derivations on triangular algebras.Linear Algebra Appl., 432, 2010, 2615-2622, Zbl 1185.47034, MR 2608180, 10.1016/j.laa.2009.12.006 .

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