Automorphisms and generalized skew derivations which are strong commutativity preserving on polynomials in prime and semiprime rings

Abstract

Let R be a prime ring of characteristic different from 2, Q _r its right Martindale quotient ring and C its extended centroid. Suppose that F, G are generalized skew derivations of R with the same associated automorphism α, and p(x ₁, …, x _n ) is a non-central polynomial over C such that

$$\left[ {F(x),\alpha (y)} \right] = G(\left[ {x,y} \right])$$

for all x, y ∈ {p(r ₁, …, r _n ): r ₁, …, r _n ∈ R}. Then there exists λ ∈ C such that F(x) = G(x) = λα(x) for all x ∈ R.