Centralizers of X-Generalized Skew Derivations on Multilinear Polynomials in Prime Rings

Abstract

Let R be a prime ring of characteristic different from 2, $Q_r$ be its right Martindale quotient ring and C be its extended centroid, G be a nonzero X-generalized skew derivation of R, and S be the set of the evaluations of a multilinear polynomial $f(x_1,\ldots ,x_n)$ over C with n non-commuting variables. Let $u,v \in R$ be such that $uG(x)x+G(x)xv=0$ for all $x\in S$. Then one of the following statements holds:

1.	$v\in C$ and there exist $a,b,c \in Q_r$ such that $G(x)=ax+bxc$ for any $x\in R$ with $(u+v)a=(u+v)b=0$;
2.	$f(x_1,\ldots ,x_n)^2$ is central-valued on R and there exists $a \in Q_r$ such that $G(x)=ax$ for all $x\in R$ with $ua+av=0$.

Keywords

X-Generalized skew derivation / Multilinear polynomial / Prime ring

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