X-generalized skew derivations with annihilating and centralizing conditions in prime rings

Abstract

Let R be a prime ring of char $(R)\ne 2$, $Q_r$ its right Martindale quotient ring and C its extended centroid, $f(x_1,\dots , x_n)$ a multilinear polynomial over C that is noncentral-valued on R and F an X-generalized skew derivation of R. If for some $0\ne a\in R$,

$$\begin{aligned} a[F(f(x_1,\dots , x_n)),f(x_1,\dots , x_n)]\in C \end{aligned}$$

for all $x_1,\dots , x_n\in R$, then one of the following holds:

• (1) There exists $\lambda \in C$ such that $F(x) = \lambda x$ for all $x \in R$;

• (2) There exist $\lambda \in C$ and $a'\in Q_r$ such that $F(x) =a'x+xa'+ \lambda x$ for all $x \in R$ and $f(x_1,\dots , x_n)^2$ is central-valued on R;

• (3) R satisfies $s_4$ and there exist $\lambda \in C$ and $a'\in Q_r$ such that $F(x) =a'x+xa'+ \lambda x$ for all $x \in R$.