Identities with skew derivations

We solve affirmatively a problem, raised by Kharchenko, on identities with compositions of skew derivations:We define the notion of trivial identities with compositions of skew derivations, which is unique in a certain sense. It is proved that if a prime ring R satisfies a nontrivial identity with c ompositions of skew derivations, then R also satisfies a generalized polyno mial identity (without skew derivations). We actually work in a more genera l context, in which higher (skew) derivations in the literature known to th e author are all covered. (C) 2000 Academic Press.