On Multiplicative (Generalized)-Derivations and Central Valued Conditions in Prime Rings

Let R be a prime ring with multiplicative (generalized)-derivations (F, f ) and (G, g) on R. This paper gives a number of central valued algebraic identities involving F and G that are equivalent to the commutativity of R under some suitable assumptions. Moreover, in order to optimize our results, we show that the assumptions taken cannot be relaxed.

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1. E. Albaş, On generalized derivations satisfying certain identities, Ukrain. Math. J. 63 (1) (2011) 596-602. DOI: 10.1007 /s11253-011-0535-7
2. A. Ali, B. Dhara, S. Khan, F. Ali, Multiplicative (generalized)-derivations and left ideals in semiprime rings, Hacettepe J. Math. Stat. 44 (6) (2015) 1293-1306.
3. M. Ashraf, N. Rehman, On derivations and commutativity in prime rings, East-West J. Math. 3 (1) (2001) 87-91.
4. M. Ashraf, A. Ali, S. Ali, Some commutativity theorems for rings with generalized derivations, Southeast Asian Bull. Math. 31 (2007) 415-421.
5. H.E. Bell, W.S. Martindale III, Centralizing mappings of semiprime rings, Canad. Math. Bull. 30 (1) (1987) 92-101. DOI: 10.4153 /CMB-1987-014-x
6. D.K. Camci, N. Aydin, On multiplicative (generalized)-derivations in semiprime rings, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 66 (1) (2017) 153-164. DOI:10.1501/Commua1 0000000784
7. B. Dhara, S. Ali, On multiplicative (generalized)-derivations in prime and semiprime rings, Aequat. Math. 86 (2013) 65-79. DOI: 10.1007/s00010-013-0205-y
8. B. Dhara, K.G. Pradhan, A note on multiplicative (generalized)-derivations with annihilator conditions, Georgian Math. J. 23 (2) (2016) 191-198. DOI:10.1515/gmj-2016-0020
9. I. Gusić, A note on generalized derivations of prime rings, Glasnik Mate. 40 (60) (2005) 47-49.
10. S. Huang, Generalized reverse derivations and commutativity of prime rings, Commun. Math. 27 (2019) 43-50. DOI:10.2478/cm-2019-0004
11. E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957) 1093-1100. DOI: 10.2307/2032686
12. N. Rehman, R.M. Al-Omary, N.M. Muthana, A note on multiplicative (generalized) (α, β)−derivations in prime rings, Annales Math. Silesianae 33 (2019) 266-275.
13. G.S. Sandhu, D. Kumar, A note on derivations and Jordan ideals of prime rings, AIMS Math. 2 (4) (2017) 580-585. DOI:10.3934/Math.2017.4.580; correction 4 (3) (2019) 684-685. OI:10.3934/Math.2019.3.684
14. G.S. Sandhu, D. Kumar, Derivations satisfying certain algebraic identities on Lie ideals, Math. Morav. 23 (2) (2019) 79-86. DOI:10.5937/MatMor1902079S