Identities involving skew Lie product and a pair of generalized derivations in prime rings with involution

Authors

DOI:

https://doi.org/10.52737/18291163-2021.13.9-1-18%20

Keywords:

Generalized derivations, involution, prime ring

Abstract

In this paper, we consider skew Lie product on an involutive ring and study several algebraic identities for it, which include generalized derivations of the ring. The results give information about the commutativity of the ring and a description of the generalized derivations.

References

A. Abbasi, M.R. Mozumder and N.A. Dar, A note on skew Lie product of prime ring with involution, Miskolc Math. Notes, 21 (2020), no. 1, pp. 203-218. https://doi.org/10.18514/mmn.2020.2644

A. Alahmadi, H. Alhazmi, S. Ali and A.N. Khan, Generalized derivations on prime rings with involution, Commun. Math. Appl., 8 (2017), no. 1, pp. 87-97.

S. Ali, N.A. Dar and M. Asci, On derivations and commutativity of prime rings with involution, Georgian Math. J., 23 (2016), no. 1, pp. 9-14. https://doi.org/10.1515/gmj-2015-0016

S. Ali, N.A. Dar and A.N. Khan, On strong commutativity preserving like maps in rings with involution, Miskolc Math. Notes, 16 (2015), no. 1, pp. 17-24. https://doi.org/10.18514/mmn.2015.1297

S. Ali, A.N. Khan and N.A. Dar, Herstein's theorem for generalized derivations in rings with involution, Hacettepe J. Math. Stat., 46 (2017), no. 6, pp. 1029-1034.

H.E. Bell and M.N. Daif, On derivations and commutativity of prime rings, Acta Math. Hungar., 66 (1995), no. 4, pp. 337-343. https://doi.org/10.1007/bf01876049

H.E. Bell and N. Rehman, Generalized derivations with commutativity and anti-commutativity conditions, Math. J. Okayama Univ., 49 (2007), pp. 139-147.

M. Brešar, On the distance of composition of two derivations to the generalized derivation, Glasgow Math. J., 33 (1991), pp. 89-93.

M. Brešar, Centralizing mappings and derivations in prime rings, J. Algebra, 156 (1993), no. 2, pp. 385-394. https://doi.org/10.1006/jabr.1993.1080

M.N. Daif, Commutativity results for semiprime rings with derivation, Internat. J. Math. Math. Sci., 21 (1998), no. 3, pp. 471-474.

N.A. Dar and S. Ali, On *-commuting mappings and derivations in rings with involution, Turkish J. Math., 40 (2016), pp. 884-894. https://doi.org/10.3906/mat-1508-61

N.A. Dar and A.N. Khan, Generalized derivations in rings with involution, Algebra Colloq., 24 (2017), no. 3, pp. 393-399. https://doi.org/10.1142/s1005386717000244

I.N. Herstein, A note on derivations, Canad. Math. Bull., 21 (1978), pp. 369-370.

M.A. Idrissi and L. Oukhtite, Some commutativity theorems for rings with involution involving generalized derivations, Asian-European J. Math., 12 (2019), no. 1, 1950001 (11 pages). https://doi.org/10.1142/s1793557119500013

T.K. Lee, Generalized derivations of left faithful rings, Commun. Algebra, 27 (1999), no. 8, pp. 4057-4073. https://doi.org/10.1080/00927879908826682

J. Ma, X.W. Xu and F.W. Niu, Strong commutativity-preserving generalized derivations on semiprime rings, Acta Math. Sinica, Eng. Ser., 24 (2008), no. 11, pp. 1835-1842. https://doi.org/10.1007/s10114-008-7445-0

A. Mamouni, L. Oukhtite, B. Nejjar and J.J. Al Jaraden, Some commutativity criteria for prime rings with differential identities on Jordan ideals, Commun. Algebra, 47 (2019), no. 1, pp. 355-361. https://doi.org/10.1080/00927872.2018.1477945

B. Nejjar, A. Kacha, A. Mamouni and L. Oukhtite, Commutativity theorems in rings with involution, Commun. Algebra, 45 (2017), no. 2, pp. 698-708. https://doi.org/10.1080/00927872.2016.1172629

L. Oukhtite, Posner's second theorem for Jordan ideals in rings with involution, Expo. Math., 29 (2011), no. 4, pp. 415-419. https://doi.org/10.1016/j.exmath.2011.07.002

G.S. Sandhu and D.K. Camci, Some results on prime rings with multiplicative derivations, Turk. J. Math., 44 (2020), no. 4, pp. 1401-1411. https://doi.org/10.3906/mat-2002-24

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Published

2021-11-04

How to Cite

Identities involving skew Lie product and a pair of generalized derivations in prime rings with involution. (2021). Armenian Journal of Mathematics, 13(9), 1-18. https://doi.org/10.52737/18291163-2021.13.9-1-18