Characterizations of $(\sigma,\tau)$-generalized Jordan derivations on prime rings

Abstract

In this paper, we characterize $(\sigma,\tau)$-generalized Jordan derivations from a ring $R$ into an $S$-bimodule $X$, where $\sigma,\tau \colon R\longrightarrow S$ are ring homomorphisms. Our result covers a known result due to Nakajima [Turkish J. Math., 30 (2006), 403-411].

Keywords

• $(\sigma,\tau)$-generalized derivation
• $(\sigma,\tau)$-generalized Jordan derivation
• Hochschild $2$-cocycle

References

1. E. Albas, On $\tau$-centralizers of semiprime rings, Siberian Math. J., 48(2) (2007), 191-196.
2. M. Bresar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc., 104(4) (1988), 1003-1006.
3. M. Bresar, Jordan derivations revisited, Math. Proc. Cambridge Philos. Soc., 139(3) (2005), 411-425.
4. M. Bresar and J. Vukman, Jordan $(\Theta,\phi)$-derivations, Glas. Mat. Ser. III, 26(46)(1-2) (1991), 13-17.
5. A. Fosner and W. Jing, A note on Jordan derivations of triangular rings, Aequationes Math., 94(2) (2020), 277-285.
6. I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc., 8 (1957), 1104-1110.
7. B. E. Johnson, Symmetric amenability and the nonexistence of Lie and Jordan derivations, Math. Proc. Cambridge Philos. Soc., 120(3) (1996), 455-473.
8. A. Nakajima, Note on generalized Jordan derivations associate with Hochschild $2$-cocycles of rings, Turkish J. Math., 30(4) (2006), 403-411.

9. A. M. Peralta and B. Russo, Automatic continuity of derivations on $C^*$-algebras and $JB^*$-triples, J. Algebra, 399 (2014), 960-977.
10. M. F. Smiley, Jordan homomorphisms onto prime rings, Proc. Amer. Math. Soc., 8 (1957), 426-429.
11. J. H. Zhang, Jordan derivations on nest algebras, Acta Math. Sinica (Chinese Ser.), 41(1) (1998), 205-212.
12. J. Zhou, Characterizations of generalized derivations associated with Hochschild $2$-cocycles and higher derivations, Quaest. Math., 39(6) (2016), 845-862.
13. A. Zivari-Kazempour, Linear maps which are $\theta$-centralizers at zero or identity products, Commun. Korean Math. Soc., 40(1) (2025), 125-136.
14. A. Zivari-Kazempour, Characterizations of $n$-Jordan multipliers on rings, J. Mahani Math. Res., 14(1) (2025), 63-72.