On Semiprime Rings with Generalized Derivations

• Journal title : Kyungpook mathematical journal
• Volume 53, Issue 4,  2013, pp.565-571
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2013.53.4.565
Title & Authors
On Semiprime Rings with Generalized Derivations
Khan, Mohd Rais; Hasnain, Mohammad Mueenul;

Abstract
In this paper, we investigate the commutativity of a semiprime ring R admitting a generalized derivation F with associated derivation D satisfying any one of the properties: (i) $\small{F(x){\circ}D(y)=[x,y]}$, (ii) $\small{D(x){\circ}F(y)=F[x,y]}$, (iii) $\small{D(x){\circ}F(y)=xy}$, (iv) $\small{F(x{\circ}y)=[F(x) y]+[D(y),x]}$, and (v) $\small{F[x,y]=F(x){\circ}y-D(y){\circ}x}$ for all x, y in some appropriate subsets of R.
Keywords
Commutators;Derivation;Ideals;Semiprime-ring;
Language
English
Cited by
References
1.
E. Albas, N. Argac, Generalized derivations of prime rings, Algebra Colloq., 11(2)(2004), 399-410.

2.
N. Argac, On prime and semiprime rings with derivations, Algebra Colloq., 13(3)(2006), 371-380.

3.
M. Ashraf, A. Ali and R. Rani, On generalized derivations of prime rings, Southeast Asian Bull. Math., 29(2005), 669-675.

4.
M. Ashraf, N. Rehman, On commutativity of rings with derivations, Results Math., 42(1-2)(2002), 3-8.

5.
M. Ashraf and N. Rehman, On derivations and commutativity in prime rings, East- West J. Math., 3(1)(2001), 87-91.

6.
M. Ashraf, N. Rehman and M. Rahman, On generalized derivations and commutativ- ity of rings, Int. J. Math., Game Theory and Algebra, 18(1)(2008), 19-24.

7.
H. E. Bell, W. S. Martindale III, Centralizing mappings of semiprime rings, Canad. Math. Bull., 30(1987), 92-101.

8.
H. E. Bell, Some commutativity results involving derivations, Trends in Theory of Rings and Modules, S. T. Rizvi and S. M. A. Zaidi (Eds), Anamaya publisher, New Delhi, India (2005).

9.
M. Bresar, On distance of the composition of two derivations to the generalized derivations, Glasgo Math. J., 33(1991), 89-93.

10.
M. N. Daif and H. E. Bell, Remarks on derivations on semiprime rings, Internat. J. Math. & Math. Sci., 15(1)(1992), 205-206.

11.
B. Hvala, Generalized derivations in rings, Comm. Algebra, 26(1998), 1147-1166.

12.
J. H. Mayne, Centralizing mappings of prime rings, Canad. Math. Bull., 27(1984), 122-126.

13.
E. C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc., 8(1957), 1093-1100.

14.
N. Rehman, On commutativity of rings with generalized derivations, Math. J. Okayama Univ., 44(2002), 43-49.