Bi-Lie n-derivations on triangular rings

Xinfeng Liang; Lingling Zhao; Xinfeng Liang; Lingling Zhao

doi:10.3934/math.2023787

AIMS Mathematics

2023, Volume 8, Issue 7: 15411-15426. doi: 10.3934/math.2023787

Previous Article Next Article

Research article

Bi-Lie n-derivations on triangular rings

Xinfeng Liang ^,,
Lingling Zhao

School of Mathematics and Big Data, Anhui University of Science and Technology, Huainan 232001, China

Received: 10 December 2022 Revised: 23 March 2023 Accepted: 16 April 2023 Published: 26 April 2023
MSC : 16W25, 15A78, 47L35

The purpose of this article is to prove that every bi-Lie n-derivation of certain triangular rings is the sum of an inner biderivation, an extremal biderivation and an additive central mapping vanishing at $ (n-1)^{th} $-commutators for both components, using the notion of maximal left ring of quotients. As a consequence, we characterize the decomposition structure of bi-Lie n-derivations on upper triangular matrix rings.
- triangular rings,
- upper triangular matrix rings,
- extremal biderivation,
- maximal left ring of quotients,
- bi-Lie n-derivation
Citation: Xinfeng Liang, Lingling Zhao. Bi-Lie n-derivations on triangular rings[J]. AIMS Mathematics, 2023, 8(7): 15411-15426. doi: 10.3934/math.2023787

Related Papers:

Abstract

The purpose of this article is to prove that every bi-Lie n-derivation of certain triangular rings is the sum of an inner biderivation, an extremal biderivation and an additive central mapping vanishing at $ (n-1)^{th} $-commutators for both components, using the notion of maximal left ring of quotients. As a consequence, we characterize the decomposition structure of bi-Lie n-derivations on upper triangular matrix rings.

References

[1]	G. Maksa, On the trace of symmetric biderivations, C. R. Math. Rep. Acad. Sci. Canada., 9 (1987), 303–308.
[2]	G. Maksa, A remark on symmetric biadditive functions having nonnegative diagonalization, Glasnik Math., 15 (1980), 279–282.
[3]	Y. Wang, Biderivations of triangular rings, Linear Multilinear Algebra, 64 (2016), 1952–1959. https://doi.org/10.1080/03081087.2015.1127887 doi: 10.1080/03081087.2015.1127887
[4]	N. M. Ghosseiri, On biderivations of upper triangular matrix rings, Linear Algebra Appl., 438 (2013), 250–260. https://doi.org/10.1016/j.laa.2012.07.039 doi: 10.1016/j.laa.2012.07.039
[5]	Y. Wang, On functional identities of degree 2 and centralizing maps in triangular rings, Oper. Matrices, 10 (2016), 485–499. https://doi.org/10.7153/oam-10-28 doi: 10.7153/oam-10-28
[6]	L. Liu, M. Y. Liu, On Jordan biderivations of triangular rings, Oper. Matrices, 15 (2021), 1417–1426. https://doi.org/10.7153/oam-2021-15-88 doi: 10.7153/oam-2021-15-88
[7]	I. N. Herstein, Lie and Jordan structures in simple associative rings, Bull. Amer. Math. Soc., 67 (1961), 517–531. https://doi.org/10.1090/S0002-9904-1961-10666-6 doi: 10.1090/S0002-9904-1961-10666-6
[8]	Y. N. Ding, 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
[9]	D. Benkovič, Generalized Lie n-derivations of triangular algebras, Commun. Algebra, 47 (2019), 5294–5302. https://doi.org/10.1080/00927872.2019.1617875 doi: 10.1080/00927872.2019.1617875
[10]	D. Eremita, Biderivations of triangular rings revisited, Bull. Malays. Math. Soc., 40 (2017), 505–527. https://doi.org/10.1007/s40840-017-0451-6 doi: 10.1007/s40840-017-0451-6
[11]	D. Benkovič, Biderivations of triangular algebras, Linear Algebra Appl., 431 (2009), 1587–1602. https://doi.org/10.1016/j.laa.2009.05.029 doi: 10.1016/j.laa.2009.05.029
[12]	X. F. Liang, D. D. Ren, F. Wei, Lie biderivations of triangular algebras, arXiv: 2002.12498.
[13]	Y. Wang, Functional identities of degree 2 in arbitrary triangular rings, Linear Algebra Appl., 479 (2015), 171–184. https://doi.org/10.1016/j.laa.2015.04.018 doi: 10.1016/j.laa.2015.04.018
[14]	D. D. Ren, X. F. Liang, Jordan biderivations on triangular algebras, Adv. Math. (China), 51 (2022), 299–312.
[15]	D. Eremita, Functional identities of degree 2 in triangular rings, Linear Algebra Appl., 438 (2013), 584–597. https://doi.org/10.1016/j.laa.2012.07.028 doi: 10.1016/j.laa.2012.07.028
[16]	Y. Utumi, On quotient rings, Osaka J. Math., 8 (1956), 1–18.
[17]	D. Eremita, 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

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)