Remarks on Generalized Derivations in Prime and Semiprime Rings
Author
Source
International Journal of Mathematics and Mathematical Sciences
Issue
Vol. 2010, Issue 2010 (31 Dec. 2010), pp.1-6, 6 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2010-12-20
Country of Publication
Egypt
No. of Pages
6
Main Subjects
Abstract EN
Let R be a ring with center Z and I a nonzero ideal of R.
An additive mapping F:R→R is called a generalized derivation of R if there exists a derivation d:R→R such that F(xy)=F(x)y+xd(y) for all x,y∈R.
In the present paper, we prove that if F([x,y])=±[x,y] for all x,y∈I or F(x∘y)=±(x∘y) for all x,y∈I, then the semiprime ring R must contains a nonzero central ideal, provided d(I)≠0.
In case R is prime ring, R must be commutative, provided d≠0.
The cases (i) F([x,y])±[x,y]∈Z and (ii) F(x∘y)±(x∘y)∈Z for all x,y∈I are also studied.
American Psychological Association (APA)
Dhara, Basudeb. 2010. Remarks on Generalized Derivations in Prime and Semiprime Rings. International Journal of Mathematics and Mathematical Sciences،Vol. 2010, no. 2010, pp.1-6.
https://search.emarefa.net/detail/BIM-487878
Modern Language Association (MLA)
Dhara, Basudeb. Remarks on Generalized Derivations in Prime and Semiprime Rings. International Journal of Mathematics and Mathematical Sciences No. 2010 (2010), pp.1-6.
https://search.emarefa.net/detail/BIM-487878
American Medical Association (AMA)
Dhara, Basudeb. Remarks on Generalized Derivations in Prime and Semiprime Rings. International Journal of Mathematics and Mathematical Sciences. 2010. Vol. 2010, no. 2010, pp.1-6.
https://search.emarefa.net/detail/BIM-487878
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-487878