On a conjecture of Vukman

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.