On a conjecture of Vukman (Q1363337)

Let \(R\) be a ring. A biadditive map \(D\colon R\times R\to R\) is called a biderivation if the maps \(x\mapsto D(x,y)\) and \(x\mapsto D(y,x)\) are derivations for every \(y\in R\). The paper considers biderivations \(D\) that are symmetric (that is, \(D(x,y)=D(y,x)\) for all \(x,y\in R\)) and such that \(f_n(x)\) is central for all \(x\in R\) and some positive integer \(n\), where \(f_n\) is defined by \(f_1(x)=D(x,x)\) and \(f_{k+1}(x)=[f_k(x),x]\). The result is that if \(R\) is a noncommutative prime ring of characteristic \(0\) or greater than \(n+2\), then \(D=0\). This answers a question posed by \textit{J. Vukman} [Aequationes Math. 40, No. 2/3, 181-189 (1990; Zbl 0716.16013)]. Reviewer's remark: \textit{W. S. Martindale III, C. R. Miers} and \textit{M. Brešar} proved that every biderivation \(D\) of a noncommutative prime ring \(R\) is of the form \(D(x,y)=\lambda[x,y]\) where \(\lambda\) is an element of the extended centroid of \(R\) [J. Algebra 161, No. 2, 342-357 (1993; Zbl 0815.16016)]. In particular, this implies that \(0\) is the only symmetric biderivation of a noncommutative prime ring of characteristic not \(2\).