A characterization of $ b $-generalized skew derivations on a Lie ideal in a prime ring

Omaima Alshanqiti; Ashutosh Pandey; Mani Shankar Pandey; Omaima Alshanqiti; Ashutosh Pandey; Mani Shankar Pandey

doi:10.3934/math.20241628

AIMS Mathematics

2024, Volume 9, Issue 12: 34184-34204. doi: 10.3934/math.20241628

Previous Article Next Article

Research article

A characterization of $ b $-generalized skew derivations on a Lie ideal in a prime ring

1.
Department of Mathematics, Umm Al-Qura University, Makkah, Saudi Arabia
2.
Department of Mathematics and Statistics, Manipal University Jaipur, Jaipur, Rajasthan 303007, India
3.
Department of Sciences, Indian Institute of Information Technology Design and Manufacturing Kurnool, Andhra Pradesh 518008, India

Received: 03 October 2024 Revised: 07 November 2024 Accepted: 21 November 2024 Published: 04 December 2024
MSC : 16N60, 16W25

This paper investigates the analysis of $ \mathrm{b} $-generalized skew derivations, denoted as $ \Delta_1 $ and $ \Delta_2 $, within a prime ring $ \mathcal{R} $ with characteristic different from 2. Here, $ \mathcal{Q}_r $ represents the right Martindale quotient ring of $ \mathcal{R} $, and $ \mathcal{C} $ denoted its extended centroid. Additionally, $ \mathcal{L} $ is a noncentral Lie ideal of $ \mathcal{R} $. Assuming $ \Delta_1 $ and $ \Delta_2 $ are nontrivial $ \mathrm{b} $-generalized skew derivations associated with the same automorphism $ \alpha $, the paper aims to explore the detailed structure of these generalized derivations that satisfy the specific equation:
$ p u \Delta_1(u) + \Delta_1(u) u q = \Delta_2(u^2), \ \text{with} \ p + q \notin \mathcal{C}, \; \; \text{for all } u \in \mathcal{L}. $
The above-studied result generalized the already existing results ^[1,2] in the literature.
- prime rings,
- Lie ideals,
- right Martindale quotient ring,
- $ \mathrm{b} $-generalized skew derivation,
- extended centroid
Citation: Omaima Alshanqiti, Ashutosh Pandey, Mani Shankar Pandey. A characterization of $ b $-generalized skew derivations on a Lie ideal in a prime ring[J]. AIMS Mathematics, 2024, 9(12): 34184-34204. doi: 10.3934/math.20241628

Related Papers:

Abstract

This paper investigates the analysis of $ \mathrm{b} $-generalized skew derivations, denoted as $ \Delta_1 $ and $ \Delta_2 $, within a prime ring $ \mathcal{R} $ with characteristic different from 2. Here, $ \mathcal{Q}_r $ represents the right Martindale quotient ring of $ \mathcal{R} $, and $ \mathcal{C} $ denoted its extended centroid. Additionally, $ \mathcal{L} $ is a noncentral Lie ideal of $ \mathcal{R} $. Assuming $ \Delta_1 $ and $ \Delta_2 $ are nontrivial $ \mathrm{b} $-generalized skew derivations associated with the same automorphism $ \alpha $, the paper aims to explore the detailed structure of these generalized derivations that satisfy the specific equation:

$ p u \Delta_1(u) + \Delta_1(u) u q = \Delta_2(u^2), \ \text{with} \ p + q \notin \mathcal{C}, \; \; \text{for all } u \in \mathcal{L}. $

The above-studied result generalized the already existing results ^[1,2] in the literature.

References

[1]	C. Gupta, On b-generalized derivations in prime rings, Rend. Circ. Mat. Palermo, Ⅱ. Ser., 72 (2023), 2703–2720. https://doi.org/10.1007/s12215-022-00817-9 doi: 10.1007/s12215-022-00817-9
[2]	B. Prajapati, C. Gupta, Composition and orthogonality of derivations with multilinear polynomials in prime rings, Rend. Circ. Mat. Palermo, Ⅱ. Ser., 69 (2020), 1279–1294. https://doi.org/10.1007/s12215-019-00473-6 doi: 10.1007/s12215-019-00473-6
[3]	V. De Filippis, F. Wei, An engel condition with x-generalized skew derivations on lie ideals, Commun. Algebra, 46 (2018), 5433–5446. https://doi.org/10.1080/00927872.2018.1469028 doi: 10.1080/00927872.2018.1469028
[4]	E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093–1100. https://doi.org/10.2307/2032686 doi: 10.2307/2032686
[5]	M. Brešar, On the distance of the composition of two derivations to the generalized derivations, Glasgow. Math. J., 33 (1991), 89–93. https://doi.org/10.1017/S0017089500008077 doi: 10.1017/S0017089500008077
[6]	V. De Filippis, Product of generalized skew derivations on Lie ideals, Commun. Algebra, 49 (2021), 2987–3009. https://doi.org/10.1080/00927872.2021.1887204 doi: 10.1080/00927872.2021.1887204
[7]	V. De Filippis, F. Wei, Centralizers of X-generalized skew derivations on multilinear polynomials in prime rings, Commun. Math. Stat., 6 (2018), 49–71. https://doi.org/10.1007/s40304-017-0125-6 doi: 10.1007/s40304-017-0125-6
[8]	V. De Filippis, G. Scudo, F. Wei, b-Generalized skew derivations on multilinear polynomials in prime rings, Polynomial Identities in Algebras, 44 (2021), 109–138. https://doi.org/10.1007/978-3-030-63111-6_7 doi: 10.1007/978-3-030-63111-6_7
[9]	A. Pandey, G. Scudo, b'-Generalized skew derivations acting as a Jordan derivation on multilinear polynomials in prime rings, Commun. Algebra, 51 (2023), 2658–2672. https://doi.org/10.1080/00927872.2023.2168915 doi: 10.1080/00927872.2023.2168915
[10]	M. S. Pandey, A. Pandey, A note on $b$-generalized skew derivations on prime rings, Advances in Ring Theory and Applications, 443 (2022), 87–101. https://doi.org/10.1007/978-3-031-50795-3_7 doi: 10.1007/978-3-031-50795-3_7
[11]	B. Dhara, G. S. Sandhu, Hypercommuting conditions of $b$-generalized skew derivations on Lie ideals in prime rings, Ricerche Mat., (2024). https://doi.org/10.1007/s11587-024-00885-2 doi: 10.1007/s11587-024-00885-2
[12]	L. Carini, G. Scudo, On Posner's theorem with b-generalized skew derivations on Lie ideals, J. Algebra Appl., 22 (2023), 2350057. https://doi.org/10.1142/S0219498823500573 doi: 10.1142/S0219498823500573
[13]	K. I. Beidar, Rings with generalized identities. 3, Vestnik Moskovskogo Universiteta Seriya i Matematika, Mekhanika, 4 (1978), 66–73.
[14]	T. K. Lee, Semiprime rings with differential identities, Bull. Inst. Math. Acad., 20 (1992), 27–38.
[15]	C. L. Chuang, T. K. Lee, Identities with a single skew derivation, J. Algebra, 288 (2005), 59–77. https://doi.org/10.1016/j.jalgebra.2003.12.032 doi: 10.1016/j.jalgebra.2003.12.032
[16]	V. De Filippis, O. M. Di Vincenzo, Vanishing derivations and centralizers of generalized derivations on multilinear polynomials, Commun. Algebra, 40 (2012), 1918–1932. https://doi.org/10.1080/00927872.2011.553859 doi: 10.1080/00927872.2011.553859
[17]	N. Argaç, V. De Filippis, Actions of generalized derivations on multilinear polynomials in prime rings, Algebra Colloq., 18 (2011), 955–964. https://doi.org/10.1142/S1005386711000836 doi: 10.1142/S1005386711000836
[18]	C. L. Chuang, The additive subgroup generated by a polynomial, Israel J. Math., 59 (1987), 98–106. https://doi.org/10.1007/BF02779669 doi: 10.1007/BF02779669
[19]	J. Bergen, I. N. Herstein, J. W. Kerr, Lie ideals and derivations of prime rings, J. Algebra, 71 (1981), 259–267. https://doi.org/10.1016/0021-8693(81)90120-4 doi: 10.1016/0021-8693(81)90120-4
[20]	V. De Filippis, B. Prajapati, S. K. Tiwari, Some generalized identities on prime rings and their application for the solution of annihilating and centralizing problems, Quaest. Math., 45 (2021), 267–305. https://doi.org/10.2989/16073606.2020.1854887 doi: 10.2989/16073606.2020.1854887
[21]	T. Erickson, W. S. Martindale, J. M. Osborn, Prime non associative algebras, Pacific J. Math., 60 (1975), 49–63.
[22]	W. S. Martindale, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12 (1969), 576–584. https://doi.org/10.1016/0021-8693(69)90029-5 doi: 10.1016/0021-8693(69)90029-5
[23]	N. Jacobson, Structure of rings, Washington: American Mathematical Society, 1956.
[24]	C. Faith, Y. Utumi, On a new proof of Litoff's theorem, Acta Mathematica Academiae Scientiarum Hungaricae, 14 (1963), 369–371. https://doi.org/10.1007/bf01895723 doi: 10.1007/bf01895723
[25]	L. Carini, V. De Filippis, G. Scudo, Identities with product of generalized skew derivations on multilinear polynomials, Commun. Algebra, 44 (2016), 3122–3138. https://doi.org/10.1080/00927872.2015.1027354 doi: 10.1080/00927872.2015.1027354
[26]	T. K. Lee, Generalized derivations of left faithful rings, Commun. Algebra, 27 (1999), 4057–4073. https://doi.org/10.1080/00927879908826682 doi: 10.1080/00927879908826682
[27]	I. N. Herstein, Topics in ring theory, Chicago: University of Chicago Press, 1969.
[28]	O. M. Di Vincenzo, On the n-th centralizer of a Lie ideal, Boll. Unione Mat. Ital., 7 (1989), 77–85.
[29]	C. Lanski, S. Montgomery, Lie structure of prime rings of characteristic 2, Pac. J. Math., 42 (1972), 117–136. https://doi.org/10.2140/pjm.1972.42.117 doi: 10.2140/pjm.1972.42.117
[30]	C. L. Chuang, Gpis having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103 (1988), 723–728. https://doi.org/10.2307/2046841 doi: 10.2307/2046841
[31]	C. J. Hou, W. M. Zhang, Q. Meng, A note on ($\alpha$, $\beta$)-derivations, Linear Algebra Appl., 432 (2010), 2600–2607. https://doi.org/10.1016/j.laa.2009.12.008 doi: 10.1016/j.laa.2009.12.008

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)