Article
Full entry | Fulltext not available
(moving wall
24 months)
Feedback
Keywords:
derivation; $b$-generalized derivation; $b$-generalized skew derivation; Lie ideal; prime ring
derivation; $b$-generalized derivation; $b$-generalized skew derivation; Lie ideal; prime ring
Summary:
Let $R$ be any noncommutative prime ring of ${\rm char}(R)\neq 2,3$, $L$ a noncentral Lie ideal of $R$ and $F$, $G$ two nonzero $b$-generalized skew derivations of $R$. Suppose that $$[F(u),u]G(u)=0$$ for all $u\in L$. Then at least one of the following conclusions holds: \item {(1)} $F(x)=\lambda x$ for all $x\in R$ and for some $\lambda \in C$, where $C$ is the extended centroid of $R$; \item {(2)} $R\subseteq M_2(K)$, the algebra of $2\times 2$ matrices over a field $K$.
References:
[1] Albaş, E., Argaç, N., Filippis, V. De: Posner's second theorem and some related annihilating conditions on Lie ideals. Filomat 32 (2018), 1285-1301. DOI 10.2298/FIL1804285A | MR 3848105 | Zbl 1499.16052
[2] Bergen, J., Herstein, I. N., Kerr, J. W.: Lie ideals and derivations of prime rings. J. Algebra 71 (1981), 259-267. DOI 10.1016/0021-8693(81)90120-4 | MR 0627439 | Zbl 0463.16023
[3] Chuang, C.-L.: GPIs having coefficients in Utumi quotient rings. Proc. Am. Math. Soc. 103 (1988), 723-728. DOI 10.1090/S0002-9939-1988-0947646-4 | MR 0947646 | Zbl 0656.16006
[4] Chuang, C.-L., Lee, T.-K.: Identities with a single skew derivation. J. Algebra 288 (2005), 59-77. DOI 10.1016/j.jalgebra.2003.12.032 | MR 2138371 | Zbl 1073.16021
[5] Dhara, B., Argac, N., Albas, E.: Vanishing derivations and co-centralizing generalized derivations on multilinear polynomials in prime rings. Commun. Algebra 44 (2016), 1905-1923. DOI 10.1080/00927872.2015.1027393 | MR 3490654 | Zbl 1344.16036
[6] Filippis, V. De: Annihilators and power values of generalized skew derivations on Lie ideals. Can. Math. Bull. 59 (2016), 258-270. DOI 10.4153/CMB-2015-077-x | MR 3492637 | Zbl 1357.16056
[7] Filippis, V. De, Vincenzo, O. M. Di: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials. Commun. Algebra 40 (2012), 1918-1932. DOI 10.1080/00927872.2011.553859 | MR 2945689 | Zbl 1258.16043
[8] Filippis, V. De, Scudo, G., El-sayiad, M. S. Tammam: An identity with generalized derivations on Lie ideals, right ideals and Banach algebras. Czech. Math. J. 62 (2012), 453-468. DOI 10.1007/s10587-012-0039-0 | MR 2990186 | Zbl 1249.16045
[9] Filippis, V. De, Wei, F.: An Engel condition with $X$-generalized skew derivations on Lie ideals. Commun. Algebra 46 (2018), 5433-5446. DOI 10.1080/00927872.2018.1469028 | MR 3923771 | Zbl 1412.16039
[10] Erickson, T. S., III, W. S. Martindale, Osborn, J. M.: Prime nonassociative algebras. Pac. J. Math. 60 (1975), 49-63. DOI 10.2140/pjm.1975.60.49 | MR 0382379 | Zbl 0355.17005
[11] Faith, C., Utumi, Y.: On a new proof of Litoff's theorem. Acta Math. Acad. Sci. Hung. 14 (1963), 369-371. DOI 10.1007/BF01895723 | MR 0155858 | Zbl 0147.28602
[12] Jacobson, N.: Structure of Rings. Colloquium Publications 37. AMS, Providence (1964). DOI 10.1090/coll/037 | MR 0222106 | Zbl 0073.02002
[13] Lanski, C.: Differential identities, Lie ideals, and Posner's theorems. Pac. J. Math. 134 (1988), 275-297. DOI 10.2140/pjm.1988.134.275 | MR 0961236 | Zbl 0614.16028
[14] Lanski, C.: An Engel condition with derivation. Proc. Am. Math. Soc. 118 (1993), 731-734. DOI 10.1090/S0002-9939-1993-1132851-9 | MR 1132851 | Zbl 0821.16037
[15] Lee, P. H., Lee, T. K.: Lie ideals of prime rings with derivations. Bull. Inst. Math., Acad. Sin. 11 (1983), 75-80. MR 0718903 | Zbl 0515.16018
[16] Liu, C.-K.: An Engel condition with $b$-generalized derivations. Linear Multilinear Algebra 65 (2017), 300-312. DOI 10.1080/03081087.2016.1183560 | MR 3577450 | Zbl 1356.16041
[17] III, W. S. Martindale: Prime rings satisfying generalized polynomial identity. J. Algebra 12 (1969), 576-584. DOI 10.1016/0021-8693(69)90029-5 | MR 0238897 | Zbl 0175.03102
[18] Posner, E. C.: Derivations in prime rings. Proc. Am. Math. Soc. 8 (1957), 1093-1100. DOI 10.1090/S0002-9939-1957-0095863-0 | MR 0095863 | Zbl 0082.03003
[19] Vukman, J.: On derivations in prime rings and Banach algebras. Proc. Am. Math. Soc. 116 (1992), 877-884. DOI 10.1090/S0002-9939-1992-1072093-8 | MR 1072093 | Zbl 0792.16034
[20] Wong, T.-L.: Derivations with power-central values on multilinear polynomials. Algebra Colloq. 3 (1996), 369-378. MR 1422975 | Zbl 0864.16031
Search
Partner of
