Derivations with annihilator conditions in prime rings. (Q2873003)

Let \(R\) be a prime ring with center \(Z(R)\), nonzero ideal \(I\), and derivation \(d\). For \(x,y\in R\) set \(x\circ y=xy+yx\). The first main result of the authors assumes that if for fixed positive integers \(n\) and \(m\), some \(a\in R\), and all \(x,y\in I\), \(a((d(x\circ y))^n-(x\circ y))^m=0\) then \(a=0\) or \(R\) is commutative. The second main result assumes further that \(\text{char\,}R\neq 2\) and for \(n\geq 1\) and \(a\in R\), as above, that \(a((d(x\circ y))^n-(x\circ y))\in Z(R)\). Then again \(a=0\) or \(R\) is commutative.