| Title:
|
Annihilators of skew derivations with Engel conditions on prime rings (English) |
| Author:
|
Pehlivan, Taylan |
| Author:
|
Albas, Emine |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
70 |
| Issue:
|
2 |
| Year:
|
2020 |
| Pages:
|
587-603 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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 $a=0$. (English) |
| Keyword:
|
prime ring |
| Keyword:
|
derivation |
| Keyword:
|
skew derivation |
| Keyword:
|
automorphism |
| MSC:
|
16W20 |
| MSC:
|
16W25 |
| idZBL:
|
07217152 |
| idMR:
|
MR4111860 |
| DOI:
|
10.21136/CMJ.2019.0412-18 |
| . |
| Date available:
|
2020-06-17T12:37:54Z |
| Last updated:
|
2022-07-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148246 |
| . |
| Reference:
|
[1] Albaş, E., Argaç, N., Filippis, V. De: Generalized derivations with Engel conditions on one-sided ideals.Commun. Algebra 36 (2008), 2063-2071. Zbl 1145.16014, MR 2418376, 10.1080/00927870801949328 |
| Reference:
|
[2] Yarbil, N. Baydar, Filippis, V. De: A quadratic differential identity with skew derivations.Commun. Algebra 46 (2018), 205-216. Zbl 1419.16025, MR 3764857, 10.1080/00927872.2017.1316853 |
| Reference:
|
[3] Beidar, K. I., III, W. S. Martindale, Mikhalev, A. V.: Rings with Generalized Identities.Pure and Applied Mathematics 196, Marcel Dekker, New York (1996). Zbl 0847.16001, MR 1368853 |
| Reference:
|
[4] Chang, J.-C.: On the identity $h(x)=af(x)+g(x)b$.Taiwanese J. Math. 7 (2003), 103-113. Zbl 1048.16018, MR 1961042, 10.11650/twjm/1500407520 |
| Reference:
|
[5] Chang, J.-C.: Generalized skew derivations with annihilating Engel conditions.Taiwanese J. Math. 12 (2008), 1641-1650. Zbl 1184.16044, MR 2449653, 10.11650/twjm/1500405076 |
| Reference:
|
[6] Chang, J.-C.: Generalized skew derivations with Engel conditions on Lie ideals.Bull. Inst. Math., Acad. Sin. (N.S.) 6 (2011), 305-320. Zbl 1275.16032, MR 2907284 |
| Reference:
|
[7] Chou, M.-C., Liu, C.-K.: Annihilators of skew derivations with Engel conditions on Lie ideals.Commun. Algebra 44 (2016), 898-911. Zbl 1343.16037, MR 3449959, 10.1080/00927872.2014.990028 |
| Reference:
|
[8] Chuang, C.-L.: Differential identities with automorphisms and antiautomorphisms I.J. Algebra 149 (1992), 371-404. Zbl 0773.16007, MR 1172436, 10.1016/0021-8693(92)90023-F |
| Reference:
|
[9] Chuang, C.-L.: Differential identities with automorphisms and antiautomorphisms II.J. Algebra 160 (1993), 130-171. Zbl 0793.16014, MR 1237081, 10.1006/jabr.1993.1181 |
| Reference:
|
[10] Chuang, C.-L., Chou, M.-C., Liu, C.-K.: Skew derivations with annihilating Engel conditions.Publ. Math. 68 (2006), 161-170. Zbl 1105.16030, MR 2213548 |
| Reference:
|
[11] 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:
|
[12] Chuang, C.-L., Liu, C.-K.: Extended Jacobson density theorem for rings with skew derivations.Commun. Algebra 35 (2007), 1391-1413. Zbl 1122.16030, MR 2313675, 10.1080/00927870601142207 |
| Reference:
|
[13] Filippis, V. De: On the annihilator of commutators with derivation in prime rings.Rend. Circ. Math. Palermo, II Ser. 49 (2000), 343-352. Zbl 0962.16017, MR 1765404, 10.1007/BF02904239 |
| Reference:
|
[14] Dhara, B., Kar, S., Pradhan, K. G.: An Engel condition of generalized derivations with annihilator on Lie ideal in prime rings.Mat. Vesn. 68 (2016), 164-174. Zbl 06750067, MR 3509647 |
| Reference:
|
[15] 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:
|
[16] Jacobson, N.: Structure of Rings.American Mathematical Society Colloquium Publications 37, AMS, Providence (1964). Zbl 0073.02002, MR 0222106, 10.1090/coll/037 |
| Reference:
|
[17] Kharchenko, V. K.: Generalized identities with automorphisms.Algebra Logic 14 (1976), 132-148 translation from Algebra Logika 14 1975 215-237. Zbl 0382.16009, MR 0399153, 10.1007/BF01668471 |
| Reference:
|
[18] Lam, T. Y.: A First Course in Noncommutative Rings.Graduate Texts in Mathematics 131, Springer, New York (1991). Zbl 0728.16001, MR 1125071, 10.1007/978-1-4684-0406-7 |
| Reference:
|
[19] Lanski, C.: An Engel condition with derivation for left ideals.Proc. Am. Math. Soc. 125 (1997), 339-345. Zbl 0869.16027, MR 1363174, 10.1090/S0002-9939-97-03673-3 |
| Reference:
|
[20] Lanski, C.: Skew derivations and Engel conditions.Commun. Algebra 42 (2014), 139-152. Zbl 1296.16050, MR 3169560, 10.1080/00927872.2012.707719 |
| Reference:
|
[21] Lee, T.-K.: Generalized derivations of left faithful rings.Commun. Algebra 27 (1999), 4057-4073. Zbl 0946.16026, MR 1700189, 10.1080/00927879908826682 |
| Reference:
|
[22] III, W. S. Martindale: Prime rings satisfying a generalized polynomial identity.J. Algebra 12 (1969), 576-584. Zbl 0175.03102, MR 0238897, 10.1016/0021-8693(69)90029-5 |
| Reference:
|
[23] 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:
|
[24] Shiue, W.-K.: Annihilators of derivations with Engel conditions on Lie ideals.Rend. Circ. Mat. Palermo (2) 52 (2003), 505-509. Zbl 1146.16307, MR 2029557, 10.1007/BF02872768 |
| Reference:
|
[25] Shiue, W.-K.: Annihilators of derivations with Engel conditions on one-sided ideals.Publ. Math. 62 (2003), 237-243. Zbl 1026.16021, MR 1956813 |
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