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 f n ( x ) ∈ Z ( R ) for all x ∈ R , where f k + 1 ( x ) = [ f k ( x ) , x ] for k ≥ 1 and f 1 ( x ) = f ( x ) , then D = 0 .