On the identity \(h(x)=af(x)+g(x)b\).
Let \(\alpha\) and \(\beta\) be automorphisms of a ring \(R\), and let \(\delta\) be an \((\alpha,\beta)\)-derivation of \(R\). The author defines the concept of a generalized \((\alpha,\beta)\)-derivation of \(R\) associated with \(\delta\) as an additive map \(f\colon R\to R\) satisfying \[ f(xy)=f(x)\alpha(y)+\beta(x)\delta(y)=\delta(x)\alpha(y)+\beta(x)f(y) \text{ for all }x,y\in R. \] The main theorem describes \((\alpha,\beta)\)-derivations \(f\), \(g\) and \(h\) of a prime ring \(R\) such that \(h(x)=af(x)+g(x)b\) for all \(x\in R\), where \(a\) and \(b\) are fixed elements. This result generalizes a result by the reviewer [J. Algebra 156, No. 2, 385-394 (1993; Zbl 0773.16017)] who treated this condition for ordinary derivations. In corollaries some other results that are known for derivations are extended to generalized \((\alpha,\beta)\)-derivations.
generalized \((\alpha,\beta)\)-derivations
scientific article; zbMATH DE number 1926032