A note on generalized derivations and left ideals of prime rings

Document Type : Research Paper

Authors

1 Department of Mathematics, Patel Memorial National College, Rajpura, India

2 Department of Mathematics, Physics and Computer Science, Sidi Mohammed Ben Abdellah University, Taza, Morocco

Abstract

Let R be a prime ring and Z(R) denotes the center of R. In this study, we expose the commutativity of R as a consequence of specific differential identities involving derivations acting on left ideals of R. Finally, we give examples that demonstrate the necessity of hypotheses taken in the theorems.

Keywords


1. A. Ali, D. Kumar and P. Miyan, On generalized derivations and commutativity
of prime and semiprime rings, Hacet. J. Math. Stat. (3) 40 (2011), 367-374.
2. R. M. Al-Omary and S. K. Nauman, Generalized derivations on prime rings
satisfying certain identities, Commun. Korean Math. Soc, (2) 36 (2021), 229-
238. DOI: 10.4134/CKMS.c200227
3. M. Ashraf and N. Rehman, On derivations and commutativity in prime rings,
East-West J. Math. (1) 3 (2001), 87-91.
4. M. Ashraf, A. Ali and S. Ali, Some commutativity theorems for rings with gen-
eralized derivations, Southeast Asian Bull. Math. 31 (2007), 415-421.
5. M. Ashraf and S. Ali, On left multipliers and the commutativity of prime rings,
Demonstr. Math. (4) XLI (2008), 763-771. DOI: 10.1515/dema-2013-0125
6. H. E. Bell, Some commutativity results involving derivations, Trends in Theory
of Rings and Modules: S. Tariq Rizvi and S.M.A. Zaidi (Eds.), 2005 Anamaya
Publ., New Delhi, India.
7. A. Boua, L. Oukhtite and A. Raji, Jordan ideals and derivations in prime near-
rings, Comment. Math. Univ. Carolin. (2) 55 (2014), 131-139.
8. H. El-Mir, A. Mamouni and L. Oukhtite, Special Mappings with Cen-
tral Values on Prime Rings, Algebra Colloq. (3) 27 (2020), 405-414. DOI:
10.1142/S1005386720000334
9. M. Hongan, A note on semiprime rings with derivations, Int. J. Math. Math.
Sci. (2) 20 (1997), 413-415. DOI: 10.1155/S0161171297000562
10. N. Jacobson, Structure theory for algebraic algebras of bounded degree, Annals
of Mathematics, (4) 46 (1945), 695-707. DOI: 10.2307/1969205
11. J. Pinter-Lucke, Commutativity conditions for rings: 1950-2005, Expo. Math.
(2) 25 (2007), 165-174. DOI: 10.1016/j.exmath.2006.07.001
12. J. H. Mayne, Centralizing mappings of prime rings, Canad. Math. Bull. (1) 27
(1984), 122-126. DOI: 10.4153/CMB-1984-018-2
13. L. Oukhtite, Posner's second theorem for Jordan ideals in rings with involution,
Expo. Math. 29 (2011), 415-419. DOI: 10.1016/j.exmath.2011.07.002
14. L. Oukhtite, A. Mamouni and M. Ashraf, Commutativity theorems for rings
with di erential identities on Jordan ideals, Comment. Math. Univ. Carolin. (4)
54 (2013), 447-4457.
15. E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. (6) 8 (1957),
1093-1100. DOI: 10.2307/2032686
16. N. Rehman, On Lie Ideals and Automorphisms in Prime Rings, Math. Notes,
(1) 107 (2020), 140-144. DOI: 10.1134/S0001434620010137
17. N. Rehman, On commutativity of rings with generalized derivations, Math. J.
Okayama Univ. 44 (2002), 43-49.
18. Rehman, N., Alnoghashi, H. M. and Hongan, M., A note on generalized deriva-
tions on prime ideals, J. Algebra Relat. Topics, (1) 10 (2022), , 159{169.
19. G. S. Sandhu and B. Davvaz, On generalized derivations and Jordan ideals
of prime rings, Rend. Circ. Mat. Palermo (2), 70 (2021), 227-233. DOI:
10.1007/s12215-020-00492-8