Generalized Skew Derivations with Invertible Values on Multilinear Polynomials

Abstract

Let R be a prime ring, f(X 1, …, X n ) a multilinear polynomial which is not central-valued on R, and G a nonzero generalized skew derivation of R. Suppose that G(f(x 1, …, x n )) is zero or invertible for all x 1, …, x n ∈ R. Then it is proved that R is either a division ring or the ring of all 2 × 2 matrices over a division ring. This result simultaneously generalizes a number of results in the literature.