International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 85612, 6 pages
doi: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 u2∈U for all u∈U. An additive function F:R→R is
called a generalized derivation if there exists a derivation d:R→R such that F(xy)=F(x)y+xd(y) holds
for all x,y∈R. In this paper, we prove that d=0 or U⊆Z(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)=x∘y or d(x)∘F(y)+x∘y=0, (4) either d(x)∘F(y)=[x,y] or d(x)∘F(y)+[x,y]=0,
(5) either d(x)∘F(y)−xy∈Z(R) or d(x)∘F(y)+xy∈Z(R), (6) either [d(x),F(y)]=[x,y] or [d(x),F(y)]+[x,y]=0,
(7) either [d(x),F(y)]=x∘y or [d(x),F(y)]+x∘y=0 for all x,y∈U.