On derivations and commutativity of prime rings with involution. (Q255123)

Let \(R\) be a prime ring with involution \(*\), and let \(Z(R)\) and \(S(R)\) denote, respectively, the center and the set of skew elements of \(R\). For \(x,y\in R\), let \([x,y]\) denote the commutator \(xy-yx\) and \(x\circ y\) denote the anticommutator \(xy+yx\).NEWLINENEWLINE It is proved that \(R\) must be commutative if \(\text{char}(R)\neq 2\), \(Z(R)\cap S(R)\neq\{0\}\) and \(R\) admits a nonzero derivation \(d\) satisfying one of the following differential identities: (i) \(d([x,x^*])=0\); (ii) \(d(x\circ x^*)=0\); (iii) \(d(xx^*)\pm xx^*=0\); (iv) \(d(xx^*)\pm x^*x=0\); (v) \(d(x)d(x^*)-xx^*=0\); (vi) \(d(x)d(x^*)-x^*x=0\).