A note on \((\sigma,\tau)\)-derivations in prime rings. (Q955909)

Let \(R\) be a ring, and let \(\sigma\) and \(\tau\) be automorphisms of \(R\). For each \(x,y\in R\), define \([x,y]_{\sigma,\tau}\) to be \(x\sigma(y)-\tau(y)x\); and let \[ C_{\sigma,\tau}=\{c\in R\mid c\sigma(x)=\tau(c)c\text{ for all }x\in R\}. \] Define a \((\sigma,\tau)\)-derivation to be an additive map \(d\colon R\to R\) such that \(d(xy)=d(x)\sigma(y)+\tau(x)d(y)\) for all \(x,y\in R\). It is proved that a prime ring \(R\) with \(\text{char}(R)\neq 2\) must be commutative if it admits a nonzero \((\sigma,\tau)\)-derivation \(d\) such that either (i) \([d(x),x]_{\sigma,\tau}\in C_{\sigma,\tau}\) for all \(x\in R\) or (ii) \(d(xy)=d(yx)\) for all \(x,y\in U\), where \(U\) is some nonzero ideal of \(R\). -- For \(\sigma=\tau=1\), \(d\) is simply a derivation, so that the condition (i) case recaptures a classic result of \textit{E. C. Posner} [Proc. Am. Math. Soc. 8, 1093-1100 (1958; Zbl 0082.03003)] and the condition (ii) case is a result of the reviewer and \textit{M. N. Daif} [Acta Math. Hung. 66, No. 4, 337-343 (1995; Zbl 0822.16033)].