Abstract
Let \(\mathcal {R}\) be a prime ring with a characteristic not equal to 2. Let \(\mathcal {U}\) and \(\mathcal {C}\) denote its Utumi quotient ring and extended centroid, respectively. Consider a non-central multilinear polynomial \(\phi (\zeta _1, \ldots , \zeta _n)\) over \(\mathcal {C}\), and let \(\textbf{G}\) be a \(\textrm{b}\)-generalized skew derivation of \(\mathcal {R}\), satisfying the identity:
The purpose of this paper is to classify all potential forms of the \(\textrm{b}\)-generalized skew derivation \(\textbf{G}.\)
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References
Argaç, N., Filippis, V. De.: Actions of generalized derivations on multilinear polynomials in prime rings. Algebra Colloquium 18, 955–964 (2011)
Beidar, K.I., Martindale, W.S., Mikhalev, A.V.: Rings with Generalized Identities. CRC Press, New York (1995)
Beidar, K.I.: Rings with generalized identities. 3. Vestnik Moskovskogo Universiteta Seriyai Matematika, Mekhanika 4, 66–73 (1978)
Brešar, M.: On the distance of the composition of two derivations to the generalized derivations. Glasg. Math. J. 33, 89–93 (1991). https://doi.org/10.1017/S0017089500008077
Brešar, M.: Centralizing mappings and derivations in prime rings. J. Algebra. 156, 385–394 (1993). https://doi.org/10.1006/jabr.1993.1080
Chuang, C.L.: Gpis having coefficients in Utumi quotient rings. Proc. Am. Math. Soc. 103(3), 723–728 (1988). https://www.ams.org/journals/proc/1988-103-03/S0002-9939-1988-0947646-4/S0002-9939-1988-0947646-4.pdf
Chuang, C.L., Lee, T.K.: Identities with a single skew derivation. J. Algebra. 288, 59–77 (2005). https://doi.org/10.1016/j.jalgebra.2003.12.032
Dhara, B.: Generalized derivations acting on multilinear polynomials in prime rings. Czechoslov. Math. J. 68, 95–119 (2018). https://doi.org/10.21136/CMJ.2017.0352-16
Erickson, T., Martindale 3rd, W.S., Osborn, J.: Prime non-associative algebras. Pac. J. Math. 60, 49–63 (1975). https://msp.org/pjm/1975/60-1/pjm-v60-n1-p06-s.pdf
Faith, C., Utumi, Y.: On a new proof of Litoff’s theorem. Acta Math. Hungarica 14, 369–371 (1963). https://doi.org/10.1007/BF01895723
De Filippis, V., Mario, O.: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials. Commun. Algebra. 40, 1918–1932 (2012). https://doi.org/10.1080/00927872.2011.553859
De Filippis, V., Wei, F.: An Engel condition with X-generalized skew derivations on lie ideals. Commun. Algebra. 46, 5433–5446 (2018). https://doi.org/10.1080/00927872.2018.1469028
De Filippis, V., Wei, F.: Centralizers of X-generalized skew derivations on multilinear polynomials in prime rings. Commun. Math. Stat. 6, 49–71 (2018). https://doi.org/10.1007/S40304-017-0125-6
De Filippis, V., Scudo, G., Wei, F.: b-Generalized skew derivations on multilinear polynomials in prime rings. Polynomial Identities in Algebras, pp. 109–138. Springer (2021). https://doi.org/10.10007/978-3-030-63111-6_7 (WEI, SCUDO, FILIP)
Jacobson, N.: Structure of Rings. American Mathematical Society, Providence, Rhode Island (1956)
Lee, T.K.: Semiprime rings with differential identities. Bull. Inst. Math., Acad. Sin. 20, 27–38 (1992). http://140.112.114.62/handle/246246/121932
Martindale 3rd, W.S.: Prime rings satisfying a generalized polynomial identity. J. Algebra 12, 576–584 (1969). https://doi.org/10.1016/0021-8693(69)90029-5
Pandey, A., Scudo, G.: b’ Generalized skew derivations acting as a Jordan derivation on multilinear polynomials in prime rings. Commun. Algebra. 51, 2658–2672 (2023). https://doi.org/10.1080/00927872.2023.2168915
Posner, E.C.: Derivations in prime rings. Proc. Am. Math. Soc. 8(6), 1093–1100 (1957). https://doi.org/10.2307/2032686
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The authors would like to express their sincere thanks to the reviewers and referees for the comments and suggestions.
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Pandey, M.S., Pandey, A. (2024). A Note on \(\textrm{b}\)-Generalized Skew Derivations on Prime Rings. In: Ali, S., Ashraf, M., De Filippis, V., Rehman, N.u. (eds) Advances in Ring Theory and Applications. WARA 2022. Springer Proceedings in Mathematics & Statistics, vol 443. Springer, Cham. https://doi.org/10.1007/978-3-031-50795-3_7
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