Document Type : Original Manuscript


1 Department of Mathematics, Government Polytechnic Kishanganj, Thakurganj, P.O. Box 855116, Kishanganj, India.

2 Department of Mathematics, Indian Institute of Technology Patna, P.O. Box 801106, Patna, India.



In this article, we characterize $\{g, h\}$-derivation on the upper triangular matrix algebra $\mathcal{T}_n(C)$ and prove that every Jordan $\{g, h\}$-derivation over $\mathcal{T}_n(C)$ is a $\{g, h\}$-derivation under a certain condition, where $C$ is a $2$-torsion free commutative ring with unity $1\neq 0$. Also, we study $\{g, h\}$-derivation and Jordan $\{g, h\}$-derivation over full matrix algebra $\mathcal{M}_n(C)$.


1. R. Alizadeh, Jordan derivations of full matrix algebras, Linear Algebra Appl., 430(1) (2009), 574–578.
2. D. Benkovič, Jordan derivations and antiderivations on triangular matrices, Linear Algebra Appl., 397 (2005), 235–244.
3. M. Brešar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc., 104(4) (1988), 1003–1006.
4. M. Brešar, Jordan {g, h}-derivations on tensor products of algebras, Linear Multilinear Algebra, 64(11) (2016), 2199–2207.
5. M. Brešar, Jordan derivations revisited, Math. Proc. Cambridge Philos. Soc., 139(3) (2005), 411–425.
6. M. Brešar and J. Vukman, On left derivations and related mappings, Proc. Amer. Math. Soc., 110(1) (1990), 7–16.
7. W. Cortes and C. Haetinger, On Jordan generalized higher derivations in rings, Turkish J. Math., 29(1) (2005), 1–11.
8. J. M. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc., 53(2) (1975), 321–324.
9. H. Ghahramani, Characterizing Jordan Derivations of Matrix Rings Through Zero Products, Math. Slovaca, 65(6) (2015), 1277–1290.
10. H. Ghahramani, Jordan derivations on block upper triangular matrix algebras, Oper. Matrices, 9(1) (2015), 181–188.
11. H. Ghahramani, M. N. Ghosseiri and L. Heidarizadeh, Linear maps on block upper triangular matrix algebras behaving like Jordan derivations through commutative zero products, Oper. Matrices, 14(1) (2020), 189–205.
12. A. Ghosh and O. Prakash, Jordan left fg; hg-derivation over some algebras, Indian J. Pure Appl. Math., 51(4) (2020), 1433–1450.
13. N. M. Ghosseiri, Jordan derivations of some classes of matrix rings, Taiwanese J. Math., 11(1) (2007), 51–62.
14. N. M. Ghosseiri, On Jordan left derivations and generalized Jordan left derivations of matrix rings, Bull. Iranian Math. Soc., 38(3) (2012), 689–698.
15. I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc., 8(6) (1957), 1104–1110.
16. M. Jiao and J. Hou, Additive maps derivable or Jordan derivable at zero point on nest algebras, Linear Algebra Appl., 432(11) (2010), 2984–2994.
17. L. Kong and J. Zhang, Jordan fg; hg-derivations on triangular algebras, Open Math., 18(1) (2020), 894–901.
18. Y. Li and D. Benkovič, Jordan generalized derivations on triangular algebras, Linear Multilinear Algebra, 59(8) (2011), 841–849.
19. F. Li and L. Wan, On P -Derivations and P -Jordan derivations of a ring, Ital. J. Pure Appl. Math., 36 (2016), 639–650.
20. F. Ma and G. Ji, Generalized Jordan derivations on triangular matrix algebras, Linear Multilinear Algebra, 55(4) (2007), 355–363.
21. A. M. Sinclair, Jordan homomorphisms and derivations on semisimple Banach algebras, Proc. Amer. Math. Soc., 24(1) (1970), 209–214.
22. J. Zhang, Jordan derivations of nest algebras, Acta Math. Sinica (Chinese Series), 41(1) (1998), 205–212.
23. J.-H. Zhang and W.-Y. Yu, Jordan derivations of triangular algebras, Linear Algebra Appl., 419(1) (2006), 251–255.