Proceedings - Mathematical Sciences

, Volume 126, Issue 3, pp 389–398

# On prime and semiprime rings with generalized derivations and non-commutative Banach algebras

• MOHD ARIF RAZA
• NADEEM UR REHMAN
Article

## Abstract

Let R be a prime ring of characteristic different from 2 and m a fixed positive integer. If R admits a generalized derivation associated with a nonzero deviation d such that [F(x),d(y)] m =[x,y] for all x,y in some appropriate subset of R, then R is commutative. Moreover, we also examine the case R is a semiprime ring. Finally, we apply the above result to Banach algebras, and we obtain a non-commutative version of the Singer–Werner theorem.

## Keywords

Banach algebras generalized derivations martindale ring of quotients prime and semiprime rings radical

## 2010 Mathematics Subject Classification

46J10 16N20 16N60 16W25

## Notes

### Acknowledgements

The authors wish to thank the referee for his/her valuable comments and suggestions.

## References

1. [1]
Beidar K I, Martindale III W S and Mikhalev A V, Rings with generalized identities, Pure and Appl. Math. (1996) (New York: Marcel Dekker Inc.) vol. 196
2. [2]
Bell H E and Daif M N, On commutativity and strong commutativity-preserving maps, Cand. Math. Bull. 37 (1994) 443–447
3. [3]
Bres̆ar M, On the distance of the composition of two derivations to the generalized derivations, Glasgow Math. J. 33 (1991) 89–93
4. [4]
Bres̆ar M and Mathieu M, Derivations mapping into the radical III, J. Funct. Anal. 133 (1995) 21–29
5. [5]
Chuang C L, GPIs having coefficients in Utumi quotient rings, Proc. Amer. Math. Soc. 103 (1988) 723–728
6. [6]
Chuang C L, Hypercentral derivations, J. Algebra 166 (1994) 34–71
7. [7]
De Filippis V, Generalized derivations in prime rings and noncommutative Banach algebras, Bull. Korean Math. Soc. 45 (2008) 621–629
8. [8]
Dhara B, Ali S and Pattanayak A, Identities with generalized derivations in semiprime rings, Demonstratio Mathematica XLVI (3) (2013) 453–460
9. [9]
Erickson T S, Martindale III W S and Osborn J M, Prime nonassociative algebras, Pacific. J. Math. 60 (1975) 49–63
10. [10]
Huang S, Derivation with Engel conditions in prime and semiprime rings, Czechoslovak Math. J. 61 (136) (2011) 1135–1140
11. [11]
Hvala B, Generalized derivations in prime rings, Comm. Algebra 26 (4) (1998) 1147–1166
12. [12]
Jacobson N, Structure of Rings, Colloquium Publications 37, Amer. Math. Soc. VII, Provindence, RI (1956)Google Scholar
13. [13]
Johnson B E and Sinclair A M, Continuity of derivations and a problem of Kaplansky, Amer. J. Math. 90 (1968) 1067–1073
14. [14]
Kharchenko V K, Differential identities of prime rings, Algebra and Logic 17 (1979) 155–168
15. [15]
Lanski C, An Engel condition with derivation, Proc. Amer. Math. Soc. 118 (1993) 731–734
16. [16]
Lee T K, Generalized derivations of left faithful rings, Comm. Algebra 27 (8) (1998) 4057–4073
17. [17]
Lee T K, Semiprime rings with differential identities, Bull. Inst. Math. Acad. Sin. 20 (1998) 27–38
18. [18]
Lin J S and Liu C K, Strong commutativity preserving maps on Lie ideals, Linear Algebra Appl. 428 (2008) 1601–1609
19. [19]
Martindale III W S, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969) 576–584
20. [20]
Mathieu M and Murphy G J, Derivations mapping into the radical, Arch. Math. 57 (1991) 469–474
21. [21]
Mathieu M and Runde V, Derivations mapping into the radical II, Bull. Lond. Math. Soc. 24 (1992) 485–487
22. [22]
Mayne J H, Centralizing mappings of prime rings, Canad. Math. Bull. 27 (1984) 122–126
23. [23]
Park K H, On derivations in noncommutative semiprime rings and Banach algebras, Bull. Korean Math. Soc. 42 (2005) 671–678
24. [24]
Posner E C, Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957) 1093–1100
25. [25]
Quadri M A, Khan M S and Rehman N, Generalized derivations and commutativity of prime rings, Indian J. Pure Appl. Math. 34 (98) (2003) 1393–1396
26. [26]
Sinclair A M, Continuous derivations on Banach algebras, Proc. Amer. Math. Soc. 20 (1969) 166–170
27. [27]
Singer I M and Wermer J, Derivations on commutative normed algebras, Math. Ann. 129 (1955) 260–264
28. [28]
Thomas M P, The image of a derivation is contained in the radical, Ann. Math. (2) 128 (3) (1988) 435–460