On derivations and commutativity in prime rings
Let \(R\) be a prime ring, \(U\) be a right ideal of \(R\), and \(d\) be a nonzero derivation of \(R\). It is shown that each of the following three conditions (i) \([d(x),d(y)] = d([y,x])\) for all \(x,y\in R\), (ii) \([d(x),d(y)] = d([x,y])\) for all \(x,y\in R\), (iii) \(\text{char\,}R\neq 2\) and \(d([x,y]) = 0\) for all \(x,y\in R\), implies that either \(R\) is commutative or \(d^ 2(U) = d(U)^ 2 = \{0\}\).
scientific article; zbMATH DE number 755487