Characterizing skew $ n $-derivations on triangular rings

He Yuan; Zhendi Gu; Jinwang Dai; He Yuan; Zhendi Gu; Jinwang Dai

doi:10.3934/era.2025334

Electronic Research Archive

2025, Volume 33, Issue 12: 7570-7583. doi: 10.3934/era.2025334

Previous Article Next Article

Research article

Characterizing skew $ n $-derivations on triangular rings

College of Mathematics and Computer, Jilin Normal University, Siping 136000, China

Received: 21 October 2025 Revised: 07 December 2025 Accepted: 11 December 2025 Published: 17 December 2025

Every ring containing a unity has a maximal right ring of quotients. In this paper, we investigate a skew $ n $-derivation $ \phi $ on a class of triangular rings and use the theory of maximal right ring of quotients to demonstrate that $ \phi $ is an extremal skew $ n $-derivation. This result not only significantly generalizes previous findings on derivations, but also forges a strong connection between the theory of derivations and that of rings of quotients, providing a tool for characterizing functional identities in various specific operator algebras.
- triangular ring,
- skew 3-derivation,
- skew $ n $-derivation,
- maximal right ring of quotients,
- extremal skew $ n $-derivation
Citation: He Yuan, Zhendi Gu, Jinwang Dai. Characterizing skew $ n $-derivations on triangular rings[J]. Electronic Research Archive, 2025, 33(12): 7570-7583. doi: 10.3934/era.2025334

Related Papers:

Abstract

Every ring containing a unity has a maximal right ring of quotients. In this paper, we investigate a skew $ n $-derivation $ \phi $ on a class of triangular rings and use the theory of maximal right ring of quotients to demonstrate that $ \phi $ is an extremal skew $ n $-derivation. This result not only significantly generalizes previous findings on derivations, but also forges a strong connection between the theory of derivations and that of rings of quotients, providing a tool for characterizing functional identities in various specific operator algebras.

References

[1]	Y. Utumi, On quotient rings, Osaka J. Math., 8 (1956), 1–18. https://doi.org/10.18910/8001
[2]	K. I. Beidar, W. S. Martindale, A. V. Mikhalev, Rings with Generalized Identities, Marcel Dekker, New York Basel Hong Kong, 1996.
[3]	W. S. Cheung, Commuting maps of triangular algebras, J. London Math. Soc., 63 (2001), 117–127. https://doi.org/10.1112/S0024610700001642 doi: 10.1112/S0024610700001642
[4]	Y. Wang, Y. Wang, Y. Q. Du, $n$-Derivations of triangular algebras, Linear Algebra Appl., 439 (2013), 463–471. https://doi.org/10.1016/j.laa.2013.03.032 doi: 10.1016/j.laa.2013.03.032
[5]	X. W. Xu, Y. Liu, W. Zhang, Skew $n$-derivations on semiprime rings, Bull. Korean Math. Soc., 50 (2012), 1863–1871. https://doi.org/10.4134/BKMS.2013.50.6.1863 doi: 10.4134/BKMS.2013.50.6.1863
[6]	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
[7]	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
[8]	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 doi: 10.1007/s40840-017-0451-6
[9]	D. Benkovič, Biderivations of triangular algebras, Linear Algebra Appl., 431 (2009), 1587–1602. https://doi.org/https://doi.org/10.1016/j.laa.2009.05.029 doi: 10.1016/j.laa.2009.05.029
[10]	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
[11]	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 doi: 10.3934/math.2023787
[12]	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
[13]	E. C. Posner, Derivations in prime rings, Proc. Am. Math. Soc., 8 (1957), 1093–1100. https://doi.org/10.1090/S0002-9939-1957-0095863-0
[14]	M. Brešar, W. S. Martindale, C. R. Miers, Centralizing maps in prime rings with involution, J. Algebra, 161 (1993), 342–357. https://doi.org/10.1006/jabr.1993.1223 doi: 10.1006/jabr.1993.1223
[15]	H. Yuan, X. K. Li, $\sigma$-Biderivations of triangular rings (in Chinese), Adv. Math., 49 (2020), 20–28.
[16]	W. S. Cheung, Mappings on Triangular Algebras, Ph.D thesis, University of Victoria, 2000.
[17]	M. Brešar, On generalized biderivations and related maps, J. Algebra, 172 (1995), 764–786. https://doi.org/10.1006/jabr.1995.1069 doi: 10.1006/jabr.1995.1069
[18]	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 doi: 10.1016/j.laa.2013.02.017

Reader Comments

Your name:*

Email:*
© 2025 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)