Identities with Generalized Skew Derivations on Lie Ideals

Abstract

Let m, n be two nonzero fixed positive integers, R a 2-torsion free prime ring with the right Martindale quotient ring Q, L a non-central Lie ideal of R, and δ a derivation of R. Suppose that α is an automorphism of R, D a skew derivation of R with the associated automorphism α, and F a generalized skew derivation of R with the associated skew derivation D. If

$$ F(x^{m+n})=F(x^m)x^{n}+x^m\delta(x^{n}) $$

is a polynomial identity for L, then either R satisfies the standard polynomial identity s ₄(x ₁, x ₂, x ₃, x ₄) of degree 4, or F is a generalized derivation of R and δ = D. Furthermore, in the latter case one of the following statements holds:

• (1) D = δ = 0 and there exists a ∈ Q such that F(x) = ax for all x ∈ R;

• (2) α is the identical mapping of R.