International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 2, Pages 263-266

On a conjecture of Vukman

Qing Deng

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.


Let R be a ring A bi-additive symmetric mapping d:R×RR is called a symmetric bi-derivation if, for any fixed yR, the mapping xD(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×RR and f:xD(x,x) be a symmetric bi-derivation and its trace, respectively. Suppose that fn(x)Z(R) for all xR, where fk+1(x)=[fk(x),x] for k1 and f1(x)=f(x), then D=0.