Geodesic
Generalized Higher Derivations on Lie Ideals of Triangular Algebras
Communications in Mathematics, Tome 25 (2017) no. 1, pp. 35-53 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 }$.
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 }$.
Classification : 15A78, 16W25, 47L35
Keywords: Admissible Lie Ideals; triangular algebra; generalized higher derivation; generalized Jordan higher derivation; generalized Jordan triple higher derivation 
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Ashraf, Mohammad; Parveen, Nazia; Wani, Bilal Ahmad. Generalized Higher Derivations on Lie Ideals of Triangular Algebras. Communications in Mathematics, Tome 25 (2017) no. 1, pp. 35-53. http://geodesic.mathdoc.fr/item/COMIM_2017_25_1_a4/

[1] Ashraf, M., Khan, A., Haetinger, C.: On $(\sigma ,\tau )$-higher derivations in prime rings. Int. Electron. J. Math., 8, 1, 2010, 65-79, | MR | Zbl

[2] Ashraf, M., Khan, A.: On generalized $(\sigma ,\tau )$-higher derivations in prime rings. SpringerPlus, 38, 2012, | MR

[3] Awtar, R.: Lie ideals and Jordan derivations of prime rings. Proc. Amer. Math. Soc., 90, 1, 1984, 9-14, | DOI | MR | Zbl

[4] Bergen, J., Herstein, I. N., Kerr, J. W.: Lie ideals and derivations of prime rings. J. Algebra, 71, 1981, 259-267, | DOI | MR | Zbl

[5] Brešar, M.: On the distance of the composition of two derivations to the generalized derivations. Glasgow Math. J., 33, 1991, 89-93, | DOI | MR | Zbl

[6] Chase, S. U.: A generalization of the ring of triangular matrices. Nagoya Math. J., 18, 1961, 13-25, | DOI | MR | Zbl

[7] Cortes, W., Haetinger, C.: On Jordan generalized higher derivations in rings. Turkish J. Math., 29, 1, 2005, 1-10, | MR | Zbl

[8] Ferrero, M., Haetinger, C.: Higher derivations and a theorem by Herstein. Quaest. Math., 25, 2, 2002, 249-257, | DOI | MR | Zbl

[9] Ferrero, M., Haetinger, C.: Higher derivations of semiprime rings. Comm. Algebra, 30, 5, 2002, 2321-2333, | DOI | MR | Zbl

[10] Haetinger, C.: Higher derivation on Lie ideals. Tend. Mat. Apl. Comput., 3, 1, 2002, 141-145, | DOI | MR

[11] Haetinger, C., Ashraf, M., Ali, S.: On Higher derivations: a survey. Int. J. Math. Game Theory Algebra, 19, 5/6, 2011, 359-379, | MR | Zbl

[12] Han, D.: Higher derivations on Lie ideals of triangular algebras. Sib. Math. J., 53, 6, 2012, 1029-1036, | DOI | MR | Zbl

[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

[14] Jing, W., Lu, S.: Generalized Jordan derivations on prime rings and standard operator algebras. Taiwanese J. Math., 7, 4, 2003, 605-613, | DOI | MR | Zbl

[15] Jung, Y. S.: Generalized Jordan triple higher derivations on prime rings. Indian J. Pure Appl. Math., 36, 9, 2005, 513-524, | MR | Zbl

[16] Lanski, C., Montgomery, S.: Lie structure of prime rings of characteristic 2. Pacific J. Math., 42, 1972, 117-136, | DOI | MR

[17] Nakajima, A.: On generalized higher derivations. Turk. J. Math., 24, 3, 2000, 295-311, | MR | Zbl

[18] Xiao, Z. H., Wei, F.: Jordan higher derivations on triangular algebras. Linear Algebra Appl., 432, 2010, 2615-2622, | DOI | MR | Zbl