Commutators and anti-commutators having automorphisms on Lie ideals in prime rings
Main Article Content
Abstract
In this manuscript, we discuss the relationship between prime rings and automorphisms satisfying differential identities involving commutators and anti-commutators on Lie ideals. In addition, we provide an example which shows that we cannot expect the same conclusion in case of semiprime rings.
Article Details
References
[1] Ali, S., Ashraf, M., Raza, M. A. and Khan, A. N., n-commuting mappings on (semi)-prime rings with applications, Comm. Algebra 5 (47) (2019), 2262–2270. Google Scholar
[2] Ashraf, M., Raza, M. A. and Parry, S., Commutators having idempotent values with automorphisms in semi-prime rings, Math. Rep. (Bucur.) 20 (70) (2018), 51—57. Google Scholar
[3] Ashraf, M., Rehman, N., On commutativity of rings with derivations, Results Math. 42 (1-2) (2002), 3–8. Google Scholar
[4] Beidar, K. I. and Bres ̆ar, M., Extended Jacobson density theorem for rings with automorphisms and derivations, Israel J. Math. 122 (2001), 317–346. Google Scholar
[5] Beidar, K. I., Martindale III, W. S., Mikhalev, A. V., Rings with Generalized Identities, Pure and Applied Mathematics, Marcel Dekker 196, New York, 1996. Google Scholar
[6] Bell, H. E. and Daif, M. N., On commutativity and strong commutativity-preserving maps, Cand. Math. Bull. 37 (1994), 443–447. Google Scholar
[7] Chuang, C. L., GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc., 103 (1988), 723–728. Google Scholar
[8] Chuang, C. L., Differential identities with automorphism and anti-automorphism-II, J. Algebra 160 (1993), 291–335. Google Scholar
[9] Daif, M. N. and Bell, H. E., Remarks on derivations on semiprime rings, Internt. J. Math. Math. Sci. 15 (1992), 205–206. Google Scholar
[10] Davvaz, B. and Raza, M. A., A note on automorphisms of Lie ideals in prime rings, Math. Slovaca 68 (5) (2018), 1223–1229. Google Scholar
[11] Deng, Q. and Ashraf, M., On strong commutativity preserving mappings, Results Math. 30 (1996), 259–263. Google Scholar
[12] Erickson, T. S., Martindale III, W., Osborn J. M., Prime nonassociative algebras, Pacfic. J. Math., bf 60 (1975), 49–63. Google Scholar
[13] Herstein, I. N., A note on derivations, Canad. Math. Bull., 21 (1978), 241–242. Google Scholar
[14] Jacobson, N., Structure of rings, Amer. Math. Soc. Colloq. Pub. 37 Rhode Island (1964). Google Scholar
[15] Lanski, C., An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993), 731–734. Google Scholar
[16] Lee, P. H. and Lee, T. K., Lie ideals of prime rings with derivations, Bull. Inst. Math. Acad. Sin., 11 (1983), 75–80. Google Scholar
[17] Martindale III, W. S., Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), 576–584. Google Scholar
[18] Mayne, J. H., Centralizing automorphisms of Lie ideals in prime rings, Canad. Math. Bull., bf 35 (1992), 510–514. Google Scholar
[19] Mayne, J. H., Centralizing mappings of prime rings, Canad. Math. Bull., 27 (1) (1984), 122–126. Google Scholar
[20] Mayne, J. H., Centralizing automorphisms of prime rings, Canad. Math. Bull., 19 (1976), 113–115. Google Scholar
[21] Posner, E. C., Derivations in prime rings, Proc. Amer. Math. Soc., 8 (1957), 1093–1100. Google Scholar
[22] Raza, M. A., On m-th-commutators and anti-commutators involving generalized derivations in prime rings, FACTA UNIVERSITATIS (NIS) Ser. Math. Inform. 34 (3) (2019), 391–398. Google Scholar
[23] Raza, M. A. and Rehman, N., An identity on automorphisms of Lie ideals in prime rings, Ann Univ Ferrara. 62 (1) (2016), 143–150. Google Scholar
[24] Rehman, N. and Raza, M. A., On m-commuting mappings with skew derivations in prime rings, St. Petersburg Math. J. 27 (4) (2016), 641–650. Google Scholar
[25] Wang, Y., Power-centralizing automorphisms of Lie ideals in prime rings, Comm. Algebra 34 (2006), 609–615. Google Scholar