Characterization of n-Lie (m₁, ...,m_n)-derivations on triangular rings

i

This article has been updated. A Correction to this article was published on .

i

The Original Article was published on

i

The Original Article was published on

i

This article has been updated

i

This article was retracted on

i

This article has been updated

Abstract

The central objective of this paper was to investigate the structure of n-Lie (m₁, ... ,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 (m₁,m₂)- 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 ≥ 3, each n-Lie (m₁, ... ,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 (m₁, ... ,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.

Keywords

n-Lie (m₁, ...,m_n)-derivation / n-Lie m-derivation / nest algebra / triangular ring

CHANGE HISTORY

A Correction to this paper has been published:
https://doi.org

The Retraction Note was updated to reflect reception of statements of disagreement from all authors.

This article has been retracted.Please see the Retraction Notice for more detail:
https://doi.org

References

[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

[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

[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

[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

[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

[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

[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

[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

[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

[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