`International Journal of Mathematics and Mathematical SciencesVolume 2007, Article ID 85612, 6 pageshttp://dx.doi.org/10.1155/2007/85612`
Research Article

## Generalized Derivations of Prime Rings

Department of Mathematics, Chuzhou University, Chuzhou 239012, China

Received 10 January 2007; Revised 8 May 2007; Accepted 19 June 2007

Academic Editor: Akbar Rhemtulla

Copyright © 2007 Huang Shuliang. 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 R be an associative prime ring, U a Lie ideal such that u2U for all uU. An additive function F:RR is called a generalized derivation if there exists a derivation d:RR such that F(xy)=F(x)y+xd(y) holds for all x,yR. In this paper, we prove that d=0 or UZ(R) if any one of the following conditions holds: (1) d(x)F(y)=0, (2) [d(x),F(y)=0], (3) either d(x)F(y)=xy or d(x)F(y)+xy=0, (4) either d(x)F(y)=[x,y] or d(x)F(y)+[x,y]=0, (5) either d(x)F(y)xyZ(R) or d(x)F(y)+xyZ(R), (6) either [d(x),F(y)]=[x,y] or [d(x),F(y)]+[x,y]=0, (7) either [d(x),F(y)]=xy or [d(x),F(y)]+xy=0 for all x,yU.