Generalized Derivations Acting on Multilinear Polynomials in Prime Rings and Banach Algebras

Abstract

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, F and G, the two nonzero generalized derivations of R, I an ideal of R and $f(x_1,\ldots ,x_n)$ a multilinear polynomial over C which is not central valued on R. If

$\begin{aligned} F(G(f(x_1,\ldots ,x_n))f(x_1,\ldots ,x_n))=0 \end{aligned}$

for all $x_1,\ldots ,x_n \in I$, then one of the followings holds: (1) there exist $a,b\in U$ such that $F(x)=ax$ and $G(x)=bx$ for all $x\in R$ with $ab=0$; (2) there exist $a,b,p\in U$ such that $F(x)=ax+xb$ and $G(x)=px$ for all $x\in R$ with $F(p)=0$ and $f(x_1,\ldots ,x_n)^2$ is central valued on R. We also obtain some related results in cases where R is a semiprime ring and Banach algebra.

Keywords

Prime ring / Derivation / Generalized derivation / Extended centroid / Utumi quotient ring / Banach algebra