International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 20, Pages 3347-3350
A note on derivations in semiprime rings
Department of Mathematics, Faculty of Education, University of Maribor, Koroška Cesta 160, Maribor 2000, Slovenia
Received 20 June 2005; Revised 2 October 2005
Copyright © 2005 Joso Vukman and Irena Kosi-Ulbl. 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.
We prove in this note the following result. Let be an integer and let be an -torsion-free semiprime ring with identity element. Suppose that there exists an additive mapping such that is fulfilled for all . In this case, is a derivation. This research is motivated by the work of Bridges and Bergen (1984). Throughout, will represent an associative ring with center . Given an integer , a ring is said to be -torsion-free if for , implies that . Recall that a ring is prime if for , implies that either or , and is semiprime in case implies that . An additive mapping is called a derivation if holds for all pairs and is called a Jordan derivation in case is fulfilled for all . Every derivation is a Jordan derivation. The converse is in general not true. A classical result of Herstein (1957) asserts that any Jordan derivation on a prime ring with characteristic different from two is a derivation. A brief proof of Herstein's result can be found in 1988 by Brešar and Vukman. Cusack (1975) generalized Herstein's result to -torsion-free semiprime rings (see also Brešar (1988) for an alternative proof). For some other results concerning derivations on prime and semiprime rings, we refer to Brešar (1989), Vukman (2005), Vukman and Kosi-Ulbl (2005).