Czechoslovak Mathematical Journal, first online, pp. 1-17


Annihilators of skew derivations with Engel conditions on prime rings

Taylan Pehlivan, Emine Albas

Received September 21, 2018.   Published online December 16, 2019.

Abstract:  Let $R$ be a noncommutative prime ring of characteristic different from 2, with its two-sided Martindale quotient ring $Q$, $C$ the extended centroid of $R$ and $a\in R$. Suppose that $\delta$ is a nonzero $\sigma$-derivation of $R$ such that $a[\delta(x^n),x^n]_k=0$ for all $x\in R$, where $\sigma$ is an automorphism of $R$, $n$ and $k$ are fixed positive integers. Then $a=0$.
Keywords:  prime ring; derivation; skew derivation; automorphism
Classification MSC:  16W20, 16W25
DOI:  10.21136/CMJ.2019.0412-18

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References:
[1] E. Albaş, N. Argaç, V. De Filippis: Generalized derivations with Engel conditions on one-sided ideals. Commun. Algebra 36 (2008), 2063-2071. DOI 10.1080/00927870801949328 | MR 2418376 | Zbl 1145.16014
[2] N. Baydar Yarbil, V. De Filippis: A quadratic differential identity with skew derivations. Commun. Algebra 46 (2018), 205-216. DOI 10.1080/00927872.2017.1316853 | MR 3764857 | Zbl 1419.16025
[3] K. I. Beidar, W. S. Martindale III, A. V. Mikhalev: Rings with Generalized Identities. Pure and Applied Mathematics 196, Marcel Dekker, New York (1996). MR 1368853 | Zbl 0847.16001
[4] J.-C. Chang: On the identity $h(x)=af(x)+g(x)b$. Taiwanese J. Math. 7 (2003), 103-113. DOI 10.11650/twjm/1500407520 | MR 1961042 | Zbl 1048.16018
[5] J.-C. Chang: Generalized skew derivations with annihilating Engel conditions. Taiwanese J. Math. 12 (2008), 1641-1650. DOI 10.11650/twjm/1500405076 | MR 2449653 | Zbl 1184.16044
[6] J.-C. Chang: Generalized skew derivations with Engel conditions on Lie ideals. Bull. Inst. Math., Acad. Sin. (N.S.) 6 (2011), 305-320. MR 2907284 | Zbl 1275.16032
[7] M.-C. Chou, C.-K. Liu: Annihilators of skew derivations with Engel conditions on Lie ideals. Commun. Algebra 44 (2016), 898-911. DOI 10.1080/00927872.2014.990028 | MR 3449959 | Zbl 1343.16037
[8] C.-L. Chuang: Differential identities with automorphisms and antiautomorphisms I. J. Algebra 149 (1992), 371-404. DOI 10.1016/0021-8693(92)90023-F | MR 1172436 | Zbl 0773.16007
[9] C.-L. Chuang: Differential identities with automorphisms and antiautomorphisms II. J. Algebra 160 (1993), 130-171. DOI 10.1006/jabr.1993.1181 | MR 1237081 | Zbl 0793.16014
[10] C.-L. Chuang, M.-C. Chou, C.-K. Liu: Skew derivations with annihilating Engel conditions. Publ. Math. 68 (2006), 161-170. MR 2213548 | Zbl 1105.16030
[11] C.-L. Chuang, T.-K. Lee: 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
[12] C.-L. Chuang, C.-K. Liu: Extended Jacobson density theorem for rings with skew derivations. Commun. Algebra 35 (2007), 1391-1413. DOI 10.1080/00927870601142207 | MR 2313675 | Zbl 1122.16030
[13] V. De Filippis: On the annihilator of commutators with derivation in prime rings. Rend. Circ. Math. Palermo, II Ser. 49 (2000), 343-352. DOI 10.1007/BF02904239 | MR 1765404 | Zbl 0962.16017
[14] B. Dhara, S. Kar, K. G. Pradhan: An Engel condition of generalized derivations with annihilator on Lie ideal in prime rings. Mat. Vesn. 68 (2016), 164-174. MR 3509647 | Zbl 06750067
[15] T. S. Erickson, W. S. Martindale III, J. M. Osborn: Prime nonassociative algebras. Pac. J. Math. 60 (1975), 49-63. DOI 10.2140/pjm.1975.60.49 | MR 0382379 | Zbl 0355.17005
[16] N. Jacobson: Structure of Rings. American Mathematical Society Colloquium Publications 37, AMS, Providence (1964). DOI 10.1090/coll/037 | MR 0222106 | Zbl 0073.02002
[17] V. K. Kharchenko: Generalized identities with automorphisms. Algebra Logic 14 (1976), 132-148 translation from Algebra Logika 14 (1975), 215-237. DOI 10.1007/BF01668471 | MR 0399153 | Zbl 0382.16009
[18] T. Y. Lam: A First Course in Noncommutative Rings. Graduate Texts in Mathematics 131, Springer, New York (1991). DOI 10.1007/978-1-4684-0406-7 | MR 1125071 | Zbl 0728.16001
[19] C. Lanski: An Engel condition with derivation for left ideals. Proc. Am. Math. Soc. 125 (1997), 339-345. DOI 10.1090/S0002-9939-97-03673-3 | MR 1363174 | Zbl 0869.16027
[20] C. Lanski: Skew derivations and Engel conditions. Commun. Algebra 42 (2014), 139-152. DOI 10.1080/00927872.2012.707719 | MR 3169560 | Zbl 1296.16050
[21] T.-K. Lee: Generalized derivations of left faithful rings. Commun. Algebra 27 (1999), 4057-4073. DOI 10.1080/00927879908826682 | MR 1700189 | Zbl 0946.16026
[22] W. S. Martindale III: Prime rings satisfying a generalized polynomial identity. J. Algebra 12 (1969), 576-584. DOI 10.1016/0021-8693(69)90029-5 | MR 0238897 | Zbl 0175.03102
[23] E. C. Posner: 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
[24] W.-K. Shiue: Annihilators of derivations with Engel conditions on Lie ideals. Rend. Circ. Mat. Palermo (2) 52 (2003), 505-509. DOI 10.1007/BF02872768 | MR 2029557 | Zbl 1146.16307
[25] W.-K. Shiue: Annihilators of derivations with Engel conditions on one-sided ideals. Publ. Math. 62 (2003), 237-243. MR 1956813 | Zbl 1026.16021

Affiliations:   Taylan Pehlivan, Emine Albas, Department of Mathematics, Ege University, C Blok, 35100 Bornova, Izmir, Turkey, e-mail: taylan_pehlivan@hotmail.com, emine.albas@ege.edu.tr


 
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