Differential identities on prime rings with involution. (English) Zbl 1395.16013
Summary: In this paper, we investigate commutativity of prime rings \(R\) with involution \(\ast\) of the second kind in which generalized derivations satisfy certain algebraic identities. Some well-known results characterizing commutativity of prime rings have been generalized. Furthermore, we provide an example to show that the restriction imposed on the involution is not superfluous.
MSC:
| 16N60 | Prime and semiprime associative rings |
| 16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |
| 16W25 | Derivations, actions of Lie algebras |
References:
| [1] | Ali, S.; Dar, N. A., On ∗-centralizing mapping in rings with involution, Georgian Math. J., 21, 1, 25-28, (2014) · Zbl 1293.16031 |
| [2] | Ali, S.; Dar, N. A.; Asci, M., On derivations and commutativity of prime rings with involution, Georgian Math. J., 23, 1, 9-14, (2016) · Zbl 1343.16033 |
| [3] | Ali, S.; Dar, N. A.; Khan, A. N., On strong commutativity preserving like maps in rings with involution, Miskolc Math. Notes, 16, 1, 17-24, (2015) · Zbl 1340.16045 |
| [4] | Ali, A.; Dhara, B.; Khan, S.; Ali, F., Multiplicative (generalized)-derivations and left ideals in semiprime rings, Hacet. J. Math. Stat., 44, 6, 1293-1306, (2015) · Zbl 1351.16041 |
| [5] | Ashraf, M.; Ali, A.; Ali, S., Some commutativity theorems for rings with generalized derivations, Southeast Asian Bull. Math., 31, 3, 415-421, (2007) · Zbl 1141.16020 |
| [6] | Ashraf, M.; Rehman, N., On derivation and commutativity in prime rings, East-West J. Math., 3, 1, 87-91, (2001) · Zbl 1009.16031 |
| [7] | Ashraf, M.; Rehman, N., On commutativity of rings with derivations, Results Math., 42, 1-2, 3-8, (2002) · Zbl 1038.16021 |
| [8] | Bell, H. E.; Daif, M. N., On commutativity and strong commutativity preserving maps, Canad. Math. Bull., 37, 443-447, (1994) · Zbl 0820.16031 |
| [9] | Bell, H. E.; Daif, M. N., On derivations and commutativity in prime rings, Acta Math. Hungar., 66, 337-343, (1995) · Zbl 0822.16033 |
| [10] | B. Nejjar, A. Kacha, L. Oukhtite and A. Mamouni, Commutativity theorems in rings with involution, Comm. Algebra (in press). · Zbl 1370.16036 |
| [11] | Oukhtite, L., On Jordan ideals and derivations in rings with involution, Comment. Math. Univ. Carolin., 51, 3, 389-395, (2010) · Zbl 1211.16037 |
| [12] | Oukhtite, L., Posner’s second theorem for Jordan ideals in rings with involution, Expo. Math., 29, 4, 415-419, (2011) · Zbl 1232.16027 |
| [13] | Oukhtite, L.; Mamouni, A., Derivations satisfying certain algebraic identities on Jordan ideals, Arab. J. Math., 1, 3, 341-346, (2012) · Zbl 1273.16038 |
| [14] | Oukhtite, L.; Mamouni, A., Generalized derivations centralizing on Jordan ideals of rings with involution, Turkish J. Math., 38, 2, 225-232, (2014) · Zbl 1300.16046 |
| [15] | Oukhtite, L.; Mamouni, A., Commutativity theorems for prime rings with generalized derivations on Jordan ideals, J. Taibah Univ. Sci., 9, 3, 314-319, (2015) |
| [16] | Posner, E. C., Derivations in prime rings, Proc. Amer. Math. Soc., 8, 1093-1100, (1957) · Zbl 0082.03003 |
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