International Journal of Nonlinear Analysis and Applications

On generalized Jordan $\ast$-derivations with associated Hochschild $\ast$-2-cocycles

Document Type : Research Paper

Authors

1 School of Mathematics, University of Damghan, Damghan, Iran

2 Department of Mathematics, Kashmar Higher Education Institute, Kashmar, Iran

Abstract

In this paper, we introduce the notions of generalized $\ast$-derivations, generalized Jordan $\ast$-derivations and Jordan triple $\ast$-derivations with the associated Hochschild $\ast$-2-cocycles and then it is proved that if $\mathcal{R}$ is a prime $\ast$-ring and $f:\mathcal{R} \rightarrow \mathcal{R}$ is a nonzero generalized $\ast$-derivation with an associated Hochschild $\ast$-2-cocycle $\beta$, then $\mathcal{R}$ is commutative. Some other results regarding generalized Jordan $\ast$-derivations are also established.

Keywords

[1] H.A. Ahmed, Right Γ-n-derivations in prime Γ-near-rings, Int. J. Nonlinnear Anal. Appl. 12 (2021), no. 2, 1653–1658.
[2] S. Ali, On generalized ∗-derivations in ∗-rings, Pales. J. Math. 1 (2012), 32–37.
[3] M. Ashraf and N. Rehman, On commutativity of rings with derivations, Results Math. 42 (2002), 3—8.
[4] M. Ashraf and M.A. Siddeeque, On ∗-n-derivations in rings with involution, Georgian Math. J. 22 (2015), no. , 9–18.
[5] M. Bre˘sar and J. Vukman, On some additive mappings in rings with involution, Aequations Math. 38 (1989), 178-185.
[6] M. Bre˘sar and B. Zalar, On the structure of Jordan ∗-derivations, Colloq. Math. 63 (1997), 163–171.
[7] M. Daif, Commutativity results for semiprime rings with derivations, Internat. J. Math. Math. Sci. 21 (1998), 471—474.
[8] H.G. Dales, Banach Algebras and Automatic Continuity, LMS Monographs 24, Clarenden Press, Oxford, 2000.
[9] B.L.M. Ferreira and B.T. Costa, ∗-Lie–Jordan-Type Maps on C ∗ -Algebras, Bull. Iranian. Math. Soc. (2021), https://doi.org/10.1007/s41980-021-00609-4.
[10] I.N. Herstein, Rings with involution, University of Chicago Press, Chicago, 1976.
[11] A. Hosseini, Φ-derivations and Commutativity of Rings and Algebras, J. Math. Ext, 16 (2022), no. 9, To appear.
[12] M. Mathieu, Where to find the image of a derivation, Functional Analysis and Operator Theory, Banach Center Publication, 30 (1994), 237–249.
[13] A. Nakajima, Note on generalized Jordan derivations associated with Hochschild 2-cocyles of rings, Turk. J. Math. 30 (2006), 403–411.
[14] S. Pedersen, Anticommuting derivations, Proc. Amer. Math. Soc. 127 (1999), 1103–1108.
[15] P. Semrl, On Jordan ∗-derivations and an application, Colloq. Math. 59 (1990), no. 2, 241–251.
[16] P. Semrl On quadratic functionals, Bull. Austral. Math. Soc. 37 (1988), no. 1, 27–28.
[17] P. Semrl, Quadratic functionals and Jordan ∗-derivation, Studia Math. 97 (1991), no. 3, 157–165.
[18] J. Vukman, On derivations in prime rings and Banach algebras, Proc. Amer. Math. Soc. 116 (1992), no. 4, 877–884.
[19] B. Zalar, Jordan ∗-derivations and quadratic functionals on octonion algebras, Commun. Algebra 22 (1994), no. 8, 2845–2859.
[20] B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carolinae. 32 (1991), no. 4, 609–614.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 1
January 2023
Pages 2155-2167
  • Receive Date: 20 March 2022
  • Accept Date: 11 June 2022