Algebraic skew derivations

Let R be a prime ring, C its extended centroid and RF (resp. Q) its left (resp. symmetric) Martindale quotient ring. Let δ be a σ-derivation of R, where σ is an automorphism of R. We show the equivalence of K-polynomials (resp. K-identities) of δ and cv-polynomials (resp. semi-invariant polynomials) in the Ore extension Q[X;σ,δ]. We prove the existence of K-polynomials of δ in certain rather general family of maps. As applications, the following are proved among other things: Consider the expression , where ai∈RF and an≠0.(1)If then either R is a GPI-ring or ϕ(x)=0 for all x∈RF.(2)If ϕ(R)⊆C then either R is commutative or ϕ(x)=0 for all x∈RF.