Characterizing $ m $-Jordan $ n $-derivations of triangular rings

He Yuan; Chuqi Jia; Lili Ma; He Yuan; Chuqi Jia; Lili Ma

doi:10.3934/math.2026115

AIMS Mathematics

2026, Volume 11, Issue 1: 2890-2906. doi: 10.3934/math.2026115

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Research article

Characterizing $ m $-Jordan $ n $-derivations of triangular rings

He Yuan ^{1
,
,},
Chuqi Jia ¹,
Lili Ma ²

1.
College of Mathematics and Computer, Jilin Normal University, Siping, Jilin 136000, China
2.
College of Science, Qiqihar University, Qiqihar, Heilongjiang 161006, China

Received: 09 November 2025 Revised: 14 January 2026 Accepted: 26 January 2026 Published: 29 January 2026
MSC : 15A78, 16W25

This paper explores the relationship between $ m $-derivations and Jordan $ n $-derivations, introducing the concept of $ m $-Jordan $ n $-derivations on triangular rings. By employing the maximal left ring of quotients, we first conduct a detailed analysis of 3-Jordan $ n $-derivations. Subsequently, using an inductive approach, we prove that under specific hypotheses, an $ m $-Jordan $ n $-derivation must be an extremal $ m $-derivation. Finally, we apply the results we obtained to upper triangular matrix rings.
- triangular ring,
- 3-Jordan $ n $-derivation,
- $ m $-Jordan $ n $-derivation,
- extremal $ m $-derivation,
- maximal left ring of quotients
Citation: He Yuan, Chuqi Jia, Lili Ma. Characterizing $ m $-Jordan $ n $-derivations of triangular rings[J]. AIMS Mathematics, 2026, 11(1): 2890-2906. doi: 10.3934/math.2026115

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Abstract

This paper explores the relationship between $ m $-derivations and Jordan $ n $-derivations, introducing the concept of $ m $-Jordan $ n $-derivations on triangular rings. By employing the maximal left ring of quotients, we first conduct a detailed analysis of 3-Jordan $ n $-derivations. Subsequently, using an inductive approach, we prove that under specific hypotheses, an $ m $-Jordan $ n $-derivation must be an extremal $ m $-derivation. Finally, we apply the results we obtained to upper triangular matrix rings.

References

[1]	I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc., 8 (1957), 1104–1110. https://doi.org/10.2307/2032688 doi: 10.2307/2032688
[2]	M. Brešar, Jordan mappings of semiprime rings, J. Algebra, 127 (1989), 218–228. https://doi.org/10.1016/0021-8693(89)90285-8 doi: 10.1016/0021-8693(89)90285-8
[3]	J. H. Zhang, W. Y. Yu, Jordan derivations of triangular algebras, Linear Algebra Appl., 419 (2006), 251–255. https://doi.org/10.1016/j.laa.2006.04.015 doi: 10.1016/j.laa.2006.04.015
[4]	X. F. Qi, Z. Guo, T. Zhang, Characterizing Jordan $n$-derivations of unital rings containing idempotents, Bull. Iran. Math. Soc., 46 (2020), 1639–1658. https://doi.org/10.1007/s41980-019-00348-7 doi: 10.1007/s41980-019-00348-7
[5]	X. F. Liang, Generalized Jordan $n$-derivations of unital algebras with idempotents, J. Math., 2021 (2021), 9997646. https://doi.org/10.1155/2021/9997646 doi: 10.1155/2021/9997646
[6]	D. Benkovič, A note on generalized Jordan $n$-derivations of unital rings, Indian J. Pure Appl. Math., 55 (2024), 623–627. https://doi.org/10.1007/s13226-023-00394-2 doi: 10.1007/s13226-023-00394-2
[7]	Y. Utumi, On quotient rings, Osaka Math. J., 8 (1956), 1–18. https://doi.org/10.18910/8001
[8]	D. Eremita, Functional identities of degree 2 in triangular rings revisited, Linear Multilinear A., 63 (2015), 534–553. https://doi.org/10.1080/03081087.2013.877012 doi: 10.1080/03081087.2013.877012
[9]	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
[10]	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
[11]	Y. Wang, Biderivations of triangular rings, Linear Multilinear A., 64 (2016), 1952–1959. https://doi.org/10.1080/03081087.2015.1127887 doi: 10.1080/03081087.2015.1127887
[12]	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
[13]	Y. Wang, Lie (Jordan) derivations of arbitrary triangular algebras, Aequat. Math., 93 (2019), 1221–1229. https://doi.org/10.1007/s00010-018-0634-8 doi: 10.1007/s00010-018-0634-8
[14]	L. Liu, M. 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
[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 doi: 10.3934/math.2023787
[16]	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

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