The central objective of this paper was to investigate the structure of $n$-Lie $(m_1, \dots, m_n)$-derivations on triangular rings from two distinct perspectives: the faithful bimodule property and the maximal left ideal quotient ring. Under certain assumptions, we established that every $2$-Lie $(m_1, m_2)$-derivation decomposes into the sum of an inner derivation, an extremal biderivation, and a bilinear central mapping. Furthermore, using mathematical induction, we proved that for $n \geq 3$, each $n$-Lie $(m_1, \dots, m_n)$-derivation can be expressed as the sum of an extremal $n$-derivation and an $n$-linear central mapping. Notably, our analysis revealed that these structural forms remain invariant regardless of whether we adopt the faithful bimodule approach or the maximal left ideal quotient ring framework. As significant corollaries, we obtained explicit structural descriptions of $n$-Lie $(m_1, \dots, m_n)$-derivations on upper triangular rings and nest algebras. Additionally, our investigation yielded several interesting extensions and generalizations of existing results in this domain.
Citation: Xinfeng Liang, Dandan Ren. Characterization of $ n $-Lie $ (m_1, \cdots, m_n) $-derivations on triangular rings[J]. Electronic Research Archive, 2026, 34(7): 4812-4845. doi: 10.3934/era.2026213
The central objective of this paper was to investigate the structure of $n$-Lie $(m_1, \dots, m_n)$-derivations on triangular rings from two distinct perspectives: the faithful bimodule property and the maximal left ideal quotient ring. Under certain assumptions, we established that every $2$-Lie $(m_1, m_2)$-derivation decomposes into the sum of an inner derivation, an extremal biderivation, and a bilinear central mapping. Furthermore, using mathematical induction, we proved that for $n \geq 3$, each $n$-Lie $(m_1, \dots, m_n)$-derivation can be expressed as the sum of an extremal $n$-derivation and an $n$-linear central mapping. Notably, our analysis revealed that these structural forms remain invariant regardless of whether we adopt the faithful bimodule approach or the maximal left ideal quotient ring framework. As significant corollaries, we obtained explicit structural descriptions of $n$-Lie $(m_1, \dots, m_n)$-derivations on upper triangular rings and nest algebras. Additionally, our investigation yielded several interesting extensions and generalizations of existing results in this domain.
| [1] | E. C. Posner, Derivations of prime rings, Proc. Am. Math. Soc., 8 (1957), 1093–1100. https://doi.org/10.4153/CMB-1983-042-2 |
| [2] |
W. S. Cheung, Lie derivations of triangular algebras, Linear Multilinear Algebra, 51 (2003), 299–310. https://doi.org/10.1080/0308108031000096993 doi: 10.1080/0308108031000096993
|
| [3] | Y. Wang, Lie n-derivations of unital algebras with idempotents, Linear Algebra Appl., 458 (2014), 512–525. https://doi.org/10.1016/j.laa.2014.06.029 |
| [4] |
Y. N. Yan, J. K. Li, Characterizations of Lie n-derivations of unital algebras with nontrivial idempotents, Filomat, 32 (2018), 4731–4754. https://doi.org/10.2298/FIL1813731D doi: 10.2298/FIL1813731D
|
| [5] | W. S. Cheung, Commuting maps of triangular algebras, J. Lond. Math. Soc., 63 (2001), 117–127. https://doi.org/10.1112/S0024610700001642 |
| [6] |
D. D. Ren, X. F. Liang, Jordan biderivations of triangular algebras, Adv. Math., 47 (2022), 299–312. https://doi.org/10.11845/sxjz.2020083b doi: 10.11845/sxjz.2020083b
|
| [7] | Y. Utumi, On quotient rings, Osaka J. Math., 8 (1956), 1–18. |
| [8] | D. Benkovič, Biderivations of triangular algebras, Linear Algebra Appl., 431 (2009), 1587–1602. https://doi.org/10.1016/j.laa.2009.05.029 |
| [9] | X. F. Liang, D. D. Ren, F. Wei, Lie biderivations of triangular algebras, preprint, arXiv: 2002.12498v1. https://doi.org/10.48550/arXiv.2002.12498 |
| [10] |
A. Jabeen, On n-Lie derivations of triangular algebras, Oper. Matrices, 16 (2022), 611–622. https://doi.org/10.7153/oam-2022-16-45 doi: 10.7153/oam-2022-16-45
|
| [11] |
Y. Wang, Functional identities of degree 2 in triangular rings revisited, Linear Multilinear Algebra, 63 (2015), 534–553. https://doi.org/10.1080/03081087.2013.877012 doi: 10.1080/03081087.2013.877012
|
| [12] |
D. Eremita, Functional identities of degree 2 in triangular rings, Linear Multilinear Algebra, 438 (2013), 584–597. https://doi.org/10.1016/j.laa.2012.07.028 doi: 10.1016/j.laa.2012.07.028
|
| [13] | D. Eremita, Biderivations of triangular rings revisited, Bull. Malays. Math. Sci. Soc., 40 (2017), 505–522. https://doi.org/10.1007/s40840-017-0451-6 |
| [14] |
D. Alghazzawi, A. Jabeen, M. A. Raza, T. Al-Sobhi, Characterization of Lie biderivations on triangular rings, Commun. Algebra, 51 (2023), 4400–4408. https://doi.org/10.1080/00927872.2023.2209809 doi: 10.1080/00927872.2023.2209809
|
| [15] | X. F. Liang, L. L. Zhao, Bi-Lie n-derivations on triangular rings, AIMS Math., 8 (2023), 15411–15426. https://doi.org/10.3934/math.2023787 |
| [16] |
X. F. Liang, H. N. Guo, Characterization of n-Lie-type derivations on triangular rings, Commun. Algebra, 52 (2024), 4368–4379. https://doi.org/10.1080/00927872.2024.2346301 doi: 10.1080/00927872.2024.2346301
|
| [17] |
D. Benkovič, Lie triple derivations of unital algebras with idempotents, Linear Multilinear Algebra, 63 (2015), 141–165. https://doi.org/10.1080/03081087.2013.851200 doi: 10.1080/03081087.2013.851200
|
| [18] | X. F. Qi, L. Q. Feng, Nonadditive commuting maps of unital rings with idempotents, Rocky Mountain J. Math., 51 (2021), 989–1000. https://doi.org/10.1216/rmj.2021.51.989 |
| [19] | M. Ashraf, M. A. Ansari, Multiplicative generalized Lie n-derivations of unital rings with idempotents, Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat., 116 (2022). https://doi.org/10.1007/s13398-022-01233-5 |
| [20] | M. A. Bahmani, D. Bennis, H. R. E. Vishki, B. Fahid, f-Biderivations and Jordan biderivations of unital algebras with idempotents, Linear Algebra Appl., 20 (2021), 2150082. https://doi.org/10.1142/S0219498821500821 |
| [21] | Y. Q. Du, Y. Wang, Biderivations of generalized matrix algebras, Linear Algebra Appl., 438 (2013), 4483–4499. https://doi.org/10.1016/j.laa.2013.02.017 |
Xinfeng Liang, Dandan Ren. Characterization of $ n $-Lie $ (m_1, \cdots, m_n) $-derivations on triangular rings[J]. Electronic Research Archive, 2026, 34(7): 4812-4845. doi: 10.3934/era.2026213
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