| Title:
|
$b$-generalized skew derivations acting on Lie ideals in prime rings (English) |
| Author:
|
Dhara, Basudeb |
| Author:
|
Singh, Kalyan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
74 |
| Issue:
|
2 |
| Year:
|
2024 |
| Pages:
|
575-597 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $K$. (English) |
| Keyword:
|
derivation |
| Keyword:
|
$b$-generalized derivation |
| Keyword:
|
$b$-generalized skew derivation |
| Keyword:
|
Lie ideal |
| Keyword:
|
prime ring |
| MSC:
|
16N60 |
| MSC:
|
16W25 |
| idZBL:
|
Zbl 07893400 |
| idMR:
|
MR4764541 |
| DOI:
|
10.21136/CMJ.2024.0507-23 |
| . |
| Date available:
|
2024-07-10T14:57:29Z |
| Last updated:
|
2026-07-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152459 |
| . |
| Reference:
|
[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. Zbl 1499.16052, MR 3848105, 10.2298/FIL1804285A |
| Reference:
|
[2] Bergen, J., Herstein, I. N., Kerr, J. W.: Lie ideals and derivations of prime rings.J. Algebra 71 (1981), 259-267. Zbl 0463.16023, MR 0627439, 10.1016/0021-8693(81)90120-4 |
| Reference:
|
[3] Chuang, C.-L.: GPIs having coefficients in Utumi quotient rings.Proc. Am. Math. Soc. 103 (1988), 723-728. Zbl 0656.16006, MR 0947646, 10.1090/S0002-9939-1988-0947646-4 |
| Reference:
|
[4] Chuang, C.-L., Lee, T.-K.: Identities with a single skew derivation.J. Algebra 288 (2005), 59-77. Zbl 1073.16021, MR 2138371, 10.1016/j.jalgebra.2003.12.032 |
| Reference:
|
[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. Zbl 1344.16036, MR 3490654, 10.1080/00927872.2015.1027393 |
| Reference:
|
[6] Filippis, V. De: Annihilators and power values of generalized skew derivations on Lie ideals.Can. Math. Bull. 59 (2016), 258-270. Zbl 1357.16056, MR 3492637, 10.4153/CMB-2015-077-x |
| Reference:
|
[7] Filippis, V. De, Vincenzo, O. M. Di: Vanishing derivations and centralizers of generalized derivations on multilinear polynomials.Commun. Algebra 40 (2012), 1918-1932. Zbl 1258.16043, MR 2945689, 10.1080/00927872.2011.553859 |
| Reference:
|
[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. Zbl 1249.16045, MR 2990186, 10.1007/s10587-012-0039-0 |
| Reference:
|
[9] Filippis, V. De, Wei, F.: An Engel condition with $X$-generalized skew derivations on Lie ideals.Commun. Algebra 46 (2018), 5433-5446. Zbl 1412.16039, MR 3923771, 10.1080/00927872.2018.1469028 |
| Reference:
|
[10] Erickson, T. S., III, W. S. Martindale, Osborn, J. M.: Prime nonassociative algebras.Pac. J. Math. 60 (1975), 49-63. Zbl 0355.17005, MR 0382379, 10.2140/pjm.1975.60.49 |
| Reference:
|
[11] Faith, C., Utumi, Y.: On a new proof of Litoff's theorem.Acta Math. Acad. Sci. Hung. 14 (1963), 369-371. Zbl 0147.28602, MR 0155858, 10.1007/BF01895723 |
| Reference:
|
[12] Jacobson, N.: Structure of Rings.Colloquium Publications 37. AMS, Providence (1964). Zbl 0073.02002, MR 0222106, 10.1090/coll/037 |
| Reference:
|
[13] Lanski, C.: Differential identities, Lie ideals, and Posner's theorems.Pac. J. Math. 134 (1988), 275-297. Zbl 0614.16028, MR 0961236, 10.2140/pjm.1988.134.275 |
| Reference:
|
[14] Lanski, C.: An Engel condition with derivation.Proc. Am. Math. Soc. 118 (1993), 731-734. Zbl 0821.16037, MR 1132851, 10.1090/S0002-9939-1993-1132851-9 |
| Reference:
|
[15] Lee, P. H., Lee, T. K.: Lie ideals of prime rings with derivations.Bull. Inst. Math., Acad. Sin. 11 (1983), 75-80. Zbl 0515.16018, MR 0718903 |
| Reference:
|
[16] Liu, C.-K.: An Engel condition with $b$-generalized derivations.Linear Multilinear Algebra 65 (2017), 300-312. Zbl 1356.16041, MR 3577450, 10.1080/03081087.2016.1183560 |
| Reference:
|
[17] III, W. S. Martindale: Prime rings satisfying generalized polynomial identity.J. Algebra 12 (1969), 576-584. Zbl 0175.03102, MR 0238897, 10.1016/0021-8693(69)90029-5 |
| Reference:
|
[18] Posner, E. C.: Derivations in prime rings.Proc. Am. Math. Soc. 8 (1957), 1093-1100. Zbl 0082.03003, MR 0095863, 10.1090/S0002-9939-1957-0095863-0 |
| Reference:
|
[19] Vukman, J.: On derivations in prime rings and Banach algebras.Proc. Am. Math. Soc. 116 (1992), 877-884. Zbl 0792.16034, MR 1072093, 10.1090/S0002-9939-1992-1072093-8 |
| Reference:
|
[20] Wong, T.-L.: Derivations with power-central values on multilinear polynomials.Algebra Colloq. 3 (1996), 369-378. Zbl 0864.16031, MR 1422975 |
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