Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter December 11, 2020

m-potent commutators of skew derivations on Lie ideals

  • Mohd Arif Raza ORCID logo EMAIL logo , Shakir Ali ORCID logo and Husain Alhazmi ORCID logo

Abstract

The purpose of this paper is to investigate the behavior of prime rings involving skew derivations with m-potent commutators on Lie ideals. In addition, we provide an example that shows that we cannot expect the same conclusion in case of semiprime rings. Also, we prove some other related results and present some open problems.

MSC 2010: 16N60; 16U80; 16W25

Acknowledgements

We thank the referees for the time given to a careful reading of the paper and making constructive comments.

References

[1] M. Ashraf, M. A. Raza and S. A. Pary, Commutators having idempotent values with automorphisms in semi-prime rings, Math. Rep. (Bucur.) 20(70) (2018), no. 1, 51–57. Search in Google Scholar

[2] K. I. Beidar and M. Brešar, Extended Jacobson density theorem for rings with derivations and automorphisms, Israel J. Math. 122 (2001), 317–346. 10.1007/BF02809906Search in Google Scholar

[3] K. I. Beidar, W. S. Martindale, III and A. V. Mikhalev, Rings with Generalized Identities, Monogr. Textb. Pure Appl. Math. 196, Marcel Dekker, New York, 1996. Search in Google Scholar

[4] L. Carini and V. De Filippis, Commutators with power central values on a Lie ideal, Pacific J. Math. 193 (2000), no. 2, 269–278. 10.2140/pjm.2000.193.269Search in Google Scholar

[5] C.-L. Chuang, Differential identities with automorphisms and antiautomorphisms. I, J. Algebra 149 (1992), no. 2, 371–404. 10.1016/0021-8693(92)90023-FSearch in Google Scholar

[6] C. L. Chuang, Differential identities with automorphisms and antiautomorphisms. II, J. Algebra 160 (1993), no. 1, 130–171. 10.1006/jabr.1993.1181Search in Google Scholar

[7] B. Davvaz and M. A. Raza, A note on automorphisms of Lie ideals in prime rings, Math. Slovaca 68 (2018), no. 5, 1223–1229. 10.1515/ms-2017-0179Search in Google Scholar

[8] V. De Filippis, Annihilators and power values of generalized skew derivations on Lie ideals, Canad. Math. Bull. 59 (2016), no. 2, 258–270. 10.4153/CMB-2015-077-xSearch in Google Scholar

[9] V. De Filippis, M. Arif Raza and N. U. Rehman, Commutators with idempotent values on multilinear polynomials in prime rings, Proc. Indian Acad. Sci. Math. Sci. 127 (2017), no. 1, 91–98. 10.1007/s12044-016-0316-1Search in Google Scholar

[10] V. De Filippis and O. M. Di Vincenzo, Generalized skew derivations on semiprime rings, Linear Multilinear Algebra 63 (2015), no. 5, 927–939. 10.1080/03081087.2014.909813Search in Google Scholar

[11] V. De Filippis and S. Huang, Power-commuting skew derivations on Lie ideals, Monatsh. Math. 177 (2015), no. 3, 363–372. 10.1007/s00605-014-0672-9Search in Google Scholar

[12] V. De Filippis, N. Rehman and M. A. Raza, Strong commutativity preserving skew derivations in semiprime rings, Bull. Malays. Math. Sci. Soc. 41 (2018), no. 4, 1819–1834. 10.1007/s40840-016-0429-9Search in Google Scholar

[13] N. J. Divinsky, On commuting automorphisms of rings, Trans. Roy. Soc. Canada Sect. III 49 (1955), 19–22. Search in Google Scholar

[14] T. S. Erickson, W. S. Martindale, III and J. M. Osborn, Prime nonassociative algebras, Pacific J. Math. 60 (1975), no. 1, 49–63. 10.2140/pjm.1975.60.49Search in Google Scholar

[15] V. K. Harčenko, Generalized identities with automorphisms, Algebra i Logika 14 (1975), no. 2, 215–237, 241. 10.1007/BF01668425Search in Google Scholar

[16] I. N. Herstein, A generalization of a theorem of Jacobson, Amer. J. Math. 73 (1951), 756–762. 10.2307/2372114Search in Google Scholar

[17] I. N. Herstein, A generalization of a theorem of Jacobson. III, Amer. J. Math. 75 (1953), 105–111. 10.2307/2372619Search in Google Scholar

[18] I. N. Herstein, The structure of a certain class of rings, Amer. J. Math. 75 (1953), 864–871. 10.2307/2372554Search in Google Scholar

[19] I. N. Herstein, A condition for the commutativity of rings, Canadian J. Math. 9 (1957), 583–586. 10.4153/CJM-1957-066-0Search in Google Scholar

[20] N. Jacobson, Structure theory for algebraic algebras of bounded degree, Ann. of Math. (2) 46 (1945), 695–707. 10.1007/978-1-4612-3692-4_33Search in Google Scholar

[21] N. Jacobson, Structure of Rings, Amer. Math. Soc. Colloq. Publ. 37, American Mathematical Society, Providence, 1964. Search in Google Scholar

[22] C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), no. 3, 731–734. 10.1090/S0002-9939-1993-1132851-9Search in Google Scholar

[23] C. Lanski and S. Montgomery, Lie structure of prime rings of characteristic 2, Pacific J. Math. 42 (1972), 117–136. 10.2140/pjm.1972.42.117Search in Google Scholar

[24] P. H. Lee and T. L. Wong, Derivations cocentralizing Lie ideals, Bull. Inst. Math. Acad. Sinica 23 (1995), no. 1, 1–5. Search in Google Scholar

[25] J. H. Mayne, Centralizing automorphisms of prime rings, Canad. Math. Bull. 19 (1976), no. 1, 113–115. 10.4153/CMB-1976-017-1Search in Google Scholar

[26] J. H. Mayne, Centralizing automorphisms of Lie ideals in prime rings, Canad. Math. Bull. 35 (1992), no. 4, 510–514. 10.4153/CMB-1992-067-0Search in Google Scholar

[27] J. Pinter-Lucke, Commutativity conditions for rings: 1950–2005, Expo. Math. 25 (2007), no. 2, 165–174. 10.1016/j.exmath.2006.07.001Search in Google Scholar

[28] M. A. Raza and N. ur Rehman, An identity on automorphisms of Lie ideals in prime rings, Ann. Univ. Ferrara Sez. VII Sci. Mat. 62 (2016), no. 1, 143–150. 10.1007/s11565-016-0240-4Search in Google Scholar

[29] N. Rehman and M. Arif Raza, On m-commuting mappings with skew derivations in prime rings (in Russian), Algebra i Analiz 27 (2015), no. 4, 74–86; translation in St. Petersburg Math. J. 27 (2016), no. 4, 641–650. 10.1090/spmj/1411Search in Google Scholar

[30] N. Rehman and V. De Filippis, On n-commuting and n-anticommuting mappings with generalized derivations in prime and semiprime rings (in Russian), Sibirsk. Mat. Zh. 52 (2011), no. 3, 655–664; translation in Sib. Math. J. 52 (2011), no. 3, 516–523. Search in Google Scholar

[31] G. Scudo and A. Z. Ansari, Generalized derivations on Lie ideals and power values on prime rings, Math. Slovaca 65 (2015), no. 5, 975–980. 10.1515/ms-2015-0066Search in Google Scholar

[32] Y. Wang, Power-centralizing automorphisms of Lie ideals in prime rings, Comm. Algebra 34 (2006), no. 2, 609–615. 10.1080/00927870500387812Search in Google Scholar

Received: 2018-05-26
Revised: 2020-03-20
Accepted: 2020-07-14
Published Online: 2020-12-11
Published in Print: 2021-12-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 24.10.2024 from https://www.degruyter.com/document/doi/10.1515/gmj-2020-2084/html
Scroll to top button