On an identity involving generalized derivations and Lie ideals of prime rings

Gurninder Singh Sandhu; Gurninder Singh Sandhu

doi:10.3934/math.2020225

AIMS Mathematics

2020, Volume 5, Issue 4: 3472-3479. doi: 10.3934/math.2020225

Previous Article Next Article

Research article

On an identity involving generalized derivations and Lie ideals of prime rings

Gurninder Singh Sandhu ^,

Department of Mathematics, Patel Memorial National College, Rajpura-140401, Punjab, India

Received: 28 December 2019 Accepted: 31 March 2020 Published: 08 April 2020
MSC : 16W25, 16N60, 16R50

Let $R$ be a prime ring, $U$ the Utumi quotient ring of $R, $ $C$ the extended centroid of $R$ and $L$ a noncentral Lie ideal of $R.$ If $R$admits a generalized derivation $F$ associated with a derivation $\delta$ of $R$ such that for some fixed integers $m, n\geq 1, $ $F([u, v])^{m} = [u, v]_{n}$ for all $u, v\in L, $ then one of the following holds true: (ⅰ) $R$ satisfies $s_{4}, $ the standard identity in four variables. (ⅱ) there exists $\lambda\in C$ such that $F(x) = \lambda x$ for all $x\in R.$ Moreover, if $n = 1, $ then $\lambda^{m} = 1$ and if $n > 1, $ then $F = 0.$
- prime ring,
- generalized derivation,
- Lie ideal,
- GPIs
Citation: Gurninder Singh Sandhu. On an identity involving generalized derivations and Lie ideals of prime rings[J]. AIMS Mathematics, 2020, 5(4): 3472-3479. doi: 10.3934/math.2020225

Related Papers:

Abstract

Let $R$ be a prime ring, $U$ the Utumi quotient ring of $R, $ $C$ the extended centroid of $R$ and $L$ a noncentral Lie ideal of $R.$ If $R$admits a generalized derivation $F$ associated with a derivation $\delta$ of $R$ such that for some fixed integers $m, n\geq 1, $ $F([u, v])^{m} = [u, v]_{n}$ for all $u, v\in L, $ then one of the following holds true: (ⅰ) $R$ satisfies $s_{4}, $ the standard identity in four variables. (ⅱ) there exists $\lambda\in C$ such that $F(x) = \lambda x$ for all $x\in R.$ Moreover, if $n = 1, $ then $\lambda^{m} = 1$ and if $n > 1, $ then $F = 0.$

References

[1]	M. Ashraf, N. Rehman, On commutativity of rings with derivations, Result. Math., 42 (2002), 3-8. doi: 10.1007/BF03323547
[2]	K. I. Beidar, Rings with generalized identities III, Moscow Univ. Math. Bull., 33 (1978), 53-58.
[3]	K. I. Beidar, W. S. Martindale III, A. V. Mikhalev, Rings with generalized identities, Pure and Applied Math., 196, New York: Marcel Dekker Inc., 1996.
[4]	J. Bergen, I. N. Herstein, J. W. Kerr, Lie ideals and derivations of prime rings, J. Algebra, 71 (1981), 259-267. doi: 10.1016/0021-8693(81)90120-4
[5]	C. L. Chuang, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103 (1988), 723-728. doi: 10.1090/S0002-9939-1988-0947646-4
[6]	M. N. Daif, H. E. Bell, Remarks on derivations on semiprime rings, International Journal of Mathematics and Mathematical Sciences, 15 (1992), 205-206. doi: 10.1155/S0161171292000255
[7]	T. S. Erickson, W. S. Martindale III, J. M. Osborn, Prime nonassociative algebras, Pacific J. Math., 60 (1975), 49-63. doi: 10.2140/pjm.1975.60.49
[8]	V. De Filippis, S. Huang, Generalized derivations on semiprime rings, Bull. Korean Math. Soc., 48 (2011), 1253-1259. doi: 10.4134/BKMS.2011.48.6.1253
[9]	S. Huang, B. Davvaz, Generalized derivations of rings and Banach algebras, Commun. Algebra, 41 (2013), 1188-1194. doi: 10.1080/00927872.2011.642043
[10]	S. Huang, N. Rehman, Generalized derivations in prime and semiprime rings, Bol. Soc. Paran. Mat., 34 (2016), 29-34. doi: 10.5269/bspm.v34i2.21774
[11]	N. Jacobson, Structure of rings, Amer. Math. Soc., Providence, RI, 1964.
[12]	V. K. Kharchenko, Differential identities of prime rings, Algebra Logic, 17 (1978), 155-168. doi: 10.1007/BF01670115
[13]	T. K. Lee, Generalized derivations of left faithful rings, Commun. Algebra, 27 (1999), 4057-4073. doi: 10.1080/00927879908826682
[14]	C. Lanski, S. Montgomery, Lie structure of prime rings of characteristic 2, Pacific J. Math., 42 (1972), 117-136. doi: 10.2140/pjm.1972.42.117
[15]	C. Lanski, An Engel condition with derivation, Proc. AMer. Math. Soc., 118 (1993), 731-734. doi: 10.1090/S0002-9939-1993-1132851-9
[16]	W. S. Martindale III, Prime rings satisfying a generalized polynomial identity, J. Algebra, 12 (1969), 576-584. doi: 10.1016/0021-8693(69)90029-5
[17]	M. A. Quadri, M. S. Khan, N. Rehman, Generalized derivations and commutativity of prime rings, Indian J. Pure Appl. Math., 34 (2003), 1393-1396.
[18]	T. L. Wong, Derivations with power values on multilinear polynomials, Algebra Colloq., 3 (1996), 369-378.

Reader Comments

Your name:*

Email:*
© 2020 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)