.svg){kind=logo}
ACTION OF $b$-GENERALIZED DERIVATIONS AS HOMOMORPHISMS AND ANTI-HOMOMORPHISMS ON RINGS
Keywords:
b-generalized derivation, prime rings, centralizers, ring of quotientsDOI:
https://doi.org/10.17654/0972555525042Abstract
We find the structure of b-generalized derivations on Martindale ring of quotients. In fact, we establish that if left b-generalized derivations act as homomorphisms and anti-homomorphisms on semiprime rings R, then R will be commutative under some specific condition. Our proof is motivated by the existing classical theory about ordinary derivation presented in [12].
Received: September 8, 2025
Accepted: October 27, 2025
References
[1] A. Ali, N. Rehman and S. Ali, On Lie ideals with derivations as homomorphisms and anti-homomorphisms, Acta Math. Hungar. 101(1-2) (2003), 79-82.
[2] A. Z. Ansari and F. Shujat, Jordan -derivations on standard operator algebras Filomat 37(1) (2023), 37-41.
[3] A. Z. Ansari, F. Shujat and A. Fallatah, Structure of (σ, ρ)–n–derivation on rings, Contemp. Math. 5(4) (2024), 5666-5678.
[4] H. E. Bell and L. C. Kappe, Rings in which derivations satisfy certain algebraic conditions, Acta Math. Hungar. 53 (1989), 339-346.
[5] M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra 156(2) (1993), 385-394.
[6] K. I. Beidar, W. S. Martindale III and A. V. Mikhalev, Rings with generalized identities, Monographs and Textbooks in Pure and Applied Mathematics, 196, Marcel Dekker, Inc, New York, 1996.
[7] V. De Filippis, A product of two generalized derivations on polynomials in prime rings, Collect. Math. 61 (2010), 303-322.
[8] H. Hvala, Generalized derivations in prime rings, Comm. Algebra 26(4) (1998), 1147-1166.
[9] M. T. Kosan and T. K. Lee, b-generalized derivations of semiprime rings having nilpotent values, J. Austral. Math. Soc. 96(3) (2014), 326-337.
[10] T. K. Lee, Generalized derivations of left faithful rings, Comm. Algebra 27(8) (1999), 4057-4073.
[11] J. H. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19 (1976), 113-115.
[12] E. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093-1100.
[13] N. Rehman, A. Z. Ansari and C. Haetinger, A note on homomorphisms and anti-homomorphisms on -ring, Thai. J. Math. 11(3) (2013), 741-750.
[14] F. Shujat and S. Khan, Left annihilators of generalized derivations on Lie ideals in prime rings, Rend. Cont. Math. Palermo 64(1) (2015), 77-81.
Downloads
Published
Issue
Section
License
Copyright (c) 2025 PUSHPA PUBLISHING HOUSE, PRAYAGRAJ, INDIA
This work is licensed under a Creative Commons Attribution 4.0 International License.
_________________________________
Attribution: Credit Pusha Publishing House as the original publisher, including title and author(s) if applicable.
Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.
No Derivatives: Modifying or creating derivative works not allowed without written permission.
Contact Pusha Publishing House for more info or permissions.
