On a theorem of J. Vukman (Q1924294)

Let \(R\) be a ring. A biadditive map \(D:R\times R\to R\) is called a biderivation if the maps \(x\mapsto D(x,y)\) and \(x\mapsto D(y,x)\) are derivations for each \(y\in R\). A biderivation \(D\) is said to be symmetric if \(D(x,y)=D(y,x)\) for all \(x,y\in R\). The main result of the paper states that if \(R\) is a noncommutative prime ring of characteristic different from 2, \(I\) is a nonzero ideal of \(R\), and \(D\) is a symmetric biderivation of \(R\) such that \([D(x,x),x]\) is central for every \(x\in I\), then \(D=0\). Reviewer's remark. The reviewer proved [J. Algebra 172, No. 3, 764-786 (1995; Zbl 0827.16024)] that every biderivation \(D:I\times I\to R\), where \(R\) is a noncommutative prime ring and \(I\) is its ideal, is of the form \(D(x,y)=\lambda[x,y]\) where \(\lambda\) is an element of the extended centroid of \(R\). In particular, this implies that actually every symmetric biderivation of \(I\times I\) into \(R\) is zero, unless \(R\) has characteristic 2.