International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 2, Pages 263-266
doi:10.1155/S0161171297000355
On a conjecture of Vukman
Department of Mathematics, Southwest China Normal University, Chongqing 630715, China
Received 27 October 1993; Revised 30 October 1995
Copyright © 1997 Qing Deng. 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 a ring A bi-additive symmetric mapping d:R×R→R is called a symmetric
bi-derivation if, for any fixed y∈R, the mapping x→D(x,y) is a derivation. The purpose of this paper
is to prove the following conjecture of Vukman.
Let R be a noncommutative prime ring with suitable characteristic restrictions, and let
D:R×R→R and f:x→D(x,x) be a symmetric bi-derivation and its trace, respectively. Suppose
that fn(x)∈Z(R) for all x∈R, where fk+1(x)=[fk(x),x] for k≥1 and f1(x)=f(x), then D=0.