b-Generalized Skew Derivations on Lie Ideals

Abstract

Let R be a non-commutative prime ring, Z(R) its center, Q its right Martindale quotient ring, C its extended centroid, $F\ne 0$ an b-generalized skew derivation of R, L a non-central Lie ideal of R, $0\ne a\in R$ and $n\ge 1$ a fixed integer. In this paper, we prove the following two results:

• 1. If R has characteristic different from 2 and 3 and $a[F(x),x]^n=0$, for all $x\in L$, then either there exists an element $\lambda \in C$, such that $F(x)=\lambda x$, for all $x\in R$ or R satisfies $s_4(x_1,\ldots ,x_4)$, the standard identity of degree 4, and there exist $\lambda \in C$ and $b\in Q$, such that $F(x)=bx+xb+\lambda x$, for all $x\in R$.

• 2. If $\mathrm{{char}}(R)=0$ or $\mathrm{{char}}(R) > n$ and $a[F(x),x]^n\in Z(R)$, for all $x\in R$, then either there exists an element $\lambda \in C$, such that $F(x)=\lambda x$, for all $x\in R$ or R satisfies $s_4(x_1,\ldots ,x_4)$.