Centralizing b-generalized derivations on multilinear polynomials

Abstract

Let $R$ be a prime ring of characteristic different from $2$ and $F$ a $b$-generalized derivation on $R$. Let $U$ be Utumi quotient ring of $R$ with extended centroid $C$ and $f(x_1,\ldots,x_n)$ be a multilinear polynomial over $C$ which is not central valued on $R$. Suppose that $d$ is a non zero derivation on $R$ such that $$d([F(f(r)), f(r)]) \in C$$ for all $r=(r_1,\ldots,r_n)\in R^n$, then one of the following holds:(1) there exist $a \in U$, $\lambda \in C$ such that $F(x)=ax+\lambda x+xa$ for all $x\in R$ and $f(x_1,\ldots,x_n)^2$ is central valued on $R$,(2) there exists $\lambda \in C$ such that $F(x)=\lambda x$ for all $x\in R$.