Generalized \((\alpha,\beta)^*\)-derivations and related mappings in semiprime \(^*\)-rings. (Q2920380)

Let \(R\) be a ring with involution \(^*\), and let \(\alpha\), \(\beta\) be endomorphisms of \(R\). An \((\alpha,\beta)^*\)-derivation is an additive map \(d\colon R\to R\) such that \(d(xy)=d(x)\alpha(y^*)+\beta(x)d(y)\) for all \(x,y\in R\); and a generalized \((\alpha,\beta)^*\)-derivation is an additive map \(F\colon R\to R\) such that \(F(xy)=F(x)\alpha(y^*)+\beta(x)d(y)\) for all \(x,y\in R\), where \(d\) is an \((\alpha,\beta)^*\)-derivation.NEWLINENEWLINE It is proved that if \(R\) is semiprime (resp. prime) and \(F\) is a generalized \((\alpha,\beta)^*\)-derivation with surjective \(\alpha\), then \(F(R)\) is central (resp. \(F=0\) or \(R\) is commutative). There are similar results for certain maps \(F\colon R\times R\to R\) known as \(\alpha^*\)-bimultipliers and generalized \((\alpha,\beta)^*\)-biderivations.