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International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 646587, 6 pages
http://dx.doi.org/10.1155/2010/646587
Research Article

Remarks on Generalized Derivations in Prime and Semiprime Rings

Department of Mathematics, Belda College, Belda, Paschim Medinipur 721424, India

Received 15 August 2010; Accepted 28 November 2010

Academic Editor: Hans Keiding

Copyright © 2010 Basudeb Dhara. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a ring with center and a nonzero ideal of . An additive mapping is called a generalized derivation of if there exists a derivation such that for all . In the present paper, we prove that if for all or for all , then the semiprime ring must contains a nonzero central ideal, provided . In case is prime ring, must be commutative, provided . The cases (i) and (ii) for all are also studied.