A note on \(\alpha\)-derivations on semiprime rings (Q2777549)

Let \(\alpha\) be an automorphism of the ring \(R\). An additive map \(d\colon R\to R\) is called an \(\alpha\)-derivation if \(d(xy)=\alpha(x)d(y)+d(x)y\) for all \(x,y\in R\). This paper extends to rings with \(\alpha\)-derivations some earlier results for rings with derivations.NEWLINENEWLINENEWLINEThe two principal results are: (a) If \(R\) is semiprime and \(d\) is a commuting \(\alpha\)-derivation, then \([x,y]d(u)=d(u)[x,y]=0\) for all \(x,y,u\in R\) and \(d(R)\) is central; (b) If \(R\) is prime with \(\text{char}(R)\neq 2\) and \(d_1\) and \(d_2\) are respectively an \(\alpha\)-derivation and a \(\beta\)-derivation such that \(\alpha\) commutes with \(d_1\) and \(d_2\) and \(\beta\) commutes with \(d_1\), then \(d_1d_2\) is an \(\alpha\beta\)-derivation only if \(d_1=0\) or \(d_2=0\). The proofs do not require the full assumption that \(\alpha\) and \(\beta\) are automorphisms of \(R\); it is sufficient to assume they are surjective endomorphisms.NEWLINENEWLINENEWLINECorollary 2.4 requires the additional hypothesis that \(d\) is commuting.