Suppose that R is a prime ring with char(R) ≠ 2 and f(ξ1, . . . , ξn) is a noncentral multilinear polynomial over C(= Z(U)), where U is the Utumi quotient ring of R. An additive mapping h : R ⟶ R is called homoderivation if h(ab) = h(a)h(b)+h(a)b+ah(b) for all a, b ∈ R. We investigate the behavior of three generalized derivations F, G, and H of R satisfying the condition
\(F\left({\xi }^{2}\right)=G\left({\xi }^{2}\right)+H\left(\xi \right)\xi +\xi H\left(\xi \right)\)
for all ξ ∈ f(R) = {f(ξ1, . . . , ξn) | ξ1, . . . , ξn ∈ R}.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Albas, “Generalized derivations on ideals of prime rings,” Miskolc Math. Notes, 14, No. 1, 3–9 (2013).
E. F. Alharfie and N. M. Muthana, “On homoderivations and commutativity of rings,” Bull. Inst. Math. Virtual Inst., 9, 301–304 (2019).
E. F. Alharfie and N. M. Muthana, “The commutativity of prime rings with homoderivations,” Int. J. Adv. Appl. Sci., 5, No. 5, 79–81 (2018).
A. Al-Kenani, A. Melaibari, and N. Muthana, “Homoderivations and commutativity of ⇤-prime rings,” East-West J. Math., 17, No. 2, 117–126 (2015).
A. Ali, N. Rehman, and S. Ali, “On Lie ideals with derivations as homomorphisms and antihomomorphisms,” Acta Math. Hungar., 101, No. 1-2, 79–82 (2003).
K. I. Beidar, W. S. Martindale, III, and A.V. Mikhalev, “Rings with generalized identities,” Pure Appl. Math., 196, Marcel Dekker, New York (1996).
C. L. Chuang, “GPIs having coefficients in Utumi quotient rings,” Proc. Amer. Math. Soc., 103, No. 3, 723–728 (1988).
V. De Filippis and O. M. Di Vincenzo, “Vanishing derivations and centralizers of generalized derivations on multilinear polynomials,” Comm. Algebra, 40, No. 6, 1918–1932 (2012).
V. De Filippis, “Generalized derivations as Jordan homomorphisms on Lie ideals and right ideals,” Acta Math. Sin. (Engl. Ser.), 25, No. 2, 1965–1974 (2009).
B. Dhara, “Generalized derivations acting on multilinear polynomials in prime rings,” Czechoslovak Math. J., 68, No. 1, 95–119 (2018).
B. Dhara, S. Sahebi, and V. Rehmani, “Generalized derivations as a generalization of Jordan homomorphisms acting on Lie ideals and right ideals,” Math. Slovaca, 65, No. 5, 963–974 (2015).
B. Dhara, “Generalized derivations acting as a homomorphism or antihomomorphism in semiprime rings,” Beitr. Algebra Geom., 53, 203–209 (2012).
M. M. El Sofy, Rings with Some Kinds of Mappings, Master’s Degree in Science Thesis, Cairo University, Branch of Fayoum, Cairo, Egypt (2000).
C. Faith and Y. Utumi, “On a new proof of Litoff’s theorem,” Acta Math. Acad. Sci. Hungar., 14, 369–371 (1963).
I. N. Herstein, “Jordan derivations of prime rings,” Proc. Amer. Math. Soc., 8, 1104–1110 (1957).
N. Jacobson, “Structure of rings,” Amer. Math. Soc. Colloq. Publ., 37 (1964).
V. K. Kharchenko, “Differential identity of prime rings,” Algebra Logic, 17, 155–168 (1978).
T. K. Lee, “Generalized derivations of left faithful rings,” Comm. Algebra, 27, No. 8, 4057–4073 (1999).
T. K. Lee, “Semiprime rings with differential identities,” Bull. Inst. Math. Acad. Sinica, 20, No. 1, 27–38 (1992).
W. S. Martindale, III and S. Wallace, , “Prime rings satisfying a generalized polynomial identity,” J. Algebra, 12, 576–584 (1969).
A. Melaibari, N. Muthana, and A. Al-Kenani, “Centrally-extended homoderivations on rings,” Gulf J. Math., 4, No. 2, 62–70 (2016).
T. L. Wong, “Derivations with power central values on multilinear polynomials,” Algebra Colloq., 3, 369–478 (1996).
Y. Wang and H. You, “Derivations as homomorphisms or antihomomorphisms on Lie ideals,” Acta Math. Sin. (Engl. Ser.), 23, No. 6, 1149–1152 (2007).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, No. 9, pp. 1178–1194, September, 2023. Ukrainian DOI: https://doi.org/10.37863/umzh.v75i9.7241.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Bera, N., Dhara, B. Jordan Homoderivation Behavior of Generalized Derivations in Prime Rings. Ukr Math J 75, 1340–1360 (2024). https://doi.org/10.1007/s11253-024-02265-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-024-02265-3