Remarks on symmetric biderivations of rings. (Q1879070)

A symmetric biderivation on a ring \(R\) is a symmetric biadditive map \(B\colon R\times R\to R\) such that \(B(xy,z)=B(x,z)y+xB(y,z)\) for all \(x,y,z\in R\). The trace \(g\) of such a map is defined by \(g(x)=B(x,x)\) for all \(x\in R\). The principal theorem states that if \(R\) is a 30-torsion-free semiprime ring and \(B\) is a symmetric biderivation such that \([[g(x),x],x]\) is central for all \(x\in R\), then \([g(x),x]=0\) for all \(x\in R\). Several other theorems assert commutativity of certain rings admitting symmetric biadditive maps or symmetric biderivations satisfying various contrived conditions.