On centrally extended Jordan derivations and related maps in rings

Year 2023, Volume: 52 Issue: 1, 23 - 35, 15.02.2023

Bharat Bhushan Gurninder S. Sandhu , Shakir Ali , Deepak Kumar

https://doi.org/10.15672/hujms.1008922

Cited By: 6

Abstract

Let $R$ be a ring and $Z(R)$ be the center of $R.$ The aim of this paper is to define the notions of centrally extended Jordan derivations and centrally extended Jordan $\ast$-derivations, and to prove some results involving these mappings. Precisely, we prove that if a $2$-torsion free noncommutative prime ring $R$ admits a centrally extended Jordan derivation (resp. centrally extended Jordan $\ast$-derivation) $\delta:R\to R$ such that
\[
[\delta(x),x]\in Z(R)~~(resp.~~[\delta(x),x^{\ast}]\in Z(R))\text{ for all }x\in R,
\]
where $'\ast'$ is an involution on $R,$ then $R$ is an order in a central simple algebra of dimension at most 4 over its center.

Keywords

Prime ring , semiprime ring , centrally extended Jordan derivation , involution , centrally extended Jordan *-derivation

References

• [1] S. Ali and N. A. Dar, On $\ast$-centralizing mappings in rings with involution, Georgian Math. J. 21 (1), 25-28, 2014.
• [2] S. Ali, A. Fo$\check{s}$ner, Maja Fo$\check{s}$ner and M. S. Khan, On generalized Jordan triple $(\alpha,\beta)^{\ast}$- derivations and related mappings, Mediterr. J. Math., 10, 1657-1668, 2013.
• [3] M. Ashraf and N. Rehman, On Jordan generalized derivations in rings, Math. J. Okayama Univ. 42, 79, 2000.
• [4] M. Ashraf, S. Ali and C. Haetinger, On derivations in rings and their applications, The Aligarh Bull. Math. 25 (2), 79-107, 2006.
• [5] K. I. Beidar, W. S. Martindale III and A. V. Mikhalev, Rings with Generalized Identities, Pure Appl. Math. 196, Marcel Dekker Inc., New York, 1996.
• [6] H. E. Bell and M. N. Daif, On centrally-extended maps on rings, Beitr. Algebra Geom. 57, 129-136, 2016.
• [7] M. Bre$\check{s}$ar, Centralizing mappings and derivations in prime rings, J. Algebra 156 (2), 385-394, 1993.
• [8] M. Bre$\check{s}$ar, Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings, Trans. Amer. Math. Soc. 335 (2), 525-546, 1993.
• [9] M. Bre$\check{s}$ar and J. Vukman, Jordan derivations on prime rings, Bull. Austral. Math. Soc. 37, 321-322, 1988.
• [10] M. Bre$\check{s}$ar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (4), 1003-1006, 1988.
• [11] M. Bre$\check{s}$ar and J. Vukman, On some additive mappings in rings with involution, Aequationes Math. 38, 178-185, 1989.
• [12] M. Bre$\check{s}$ar and B. Zalar, On the structure of Jordan $\ast$-derivations, Colloq. Math. 63, 163-171, 1992.
• [13] N. A. Dar and S. Ali, On the structure of generalized Jordan $\ast$-derivations of prime rings, Commun. Algeb. 49 (4), 1422-1430, 2021.
• [14] S. F. El-Deken and H. Nabiel, Centrally-extended generalized $\ast$-derivations on rings with involution, Beitr Algebra Geom. 60, 217-224, 2019.
• [15] S. F. El-Deken and M. M. El-Soufi, On centrally extended reverse and generalized reverse derivations, Indian J. Pure Appl. Math., 51 (3), 1165-1180, 2020.
• [16] N. Divinsky, On commuting automorphisms of rings, Trans. Royal Soc. Can. Sec. III 3 (49), 19-22, 1955.
• [17] I. N. Herstein, Rings with involution, University of Chicago Press, Chicago, 1976.
• [18] I. N. Herstein, Jordan derivations of prime rings, Proc. Amer. Math. Soc. 8, 1104- 1119, 1957.
• [19] C. Lanski, Lie structure in semi-prime rings with involution, Commun. Algeb. 4 (8), 731-746, 1976.
• [20] T. K. Lee and P. H. Lee, Derivations centralizing symmetric or skew elements, Bull. Inst. Math. Acad. Sini. 14 (3), 249-256, 1986.
• [21] T. K. Lee and Y. Zhou, Jordan $\ast$-derivations of prime rings, J. Algebra Appl. 13 (4), 1350126 (9 pages), 2014.
• [22] T. K. Lee, T. L. Wong and Y. Zhou, The structure of Jordan $\ast$-derivations of prime rings, Linear Multi. Algeb. 63 (2), 411-422, 2015.
• [23] N. Muthana and Z. Alkhmisi, On centrally-extended multiplicative (generalized)-($\alpha,$ $\beta$)-derivations in semiprime rings, Hacettepe J. Math. Stat. 49 (2), 578-585, 2020.
• [24] E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8, 1093-1100, 1957.
• [25] P. $\check{S}$emrl, On Jordan $\ast$-derivations and an application, Colloq. Math. 59, 241-251, 1990.
• [26] P. $\check{S}$emrl, Quadratic functionals and Jordan $\ast$-derivations, Stud. Math. 97, 157-165, 1991.
• [27] B. Zalar, On centralisers of semiprime rings, Comment. Math. Univ. Carolina. 32 (4), 609-614, 1991.