Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

HOME ALL ARTICLES View

Bull. Korean Math. Soc. 2018; 55(2): 573-586

Online first article January 9, 2018      Printed March 30, 2018

https://doi.org/10.4134/BKMS.b170153

Copyright © The Korean Mathematical Society.

$b$-generalized derivations on multilinear polynomials in prime rings

Basudeb Dhara

Belda College

Abstract

Let $R$ be a noncommutative prime ring of characteristic different from $2$, $Q$ be its maximal right ring of quotients and $C$ be its extended centroid. Suppose that $f(x_1,\ldots,x_n)$ be a noncentral multilinear polynomial over $C$, $b\in Q$, $F$ a $b$-generalized derivation of $R$ and $d$ is a nonzero derivation of $R$ such that $$d([F(f(r)),f(r)])=0$$ for all $r=(r_1,\ldots,r_n)\in R^n$. 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 $p\in Q$ such that $F(x)=\lambda x+px+xp$ for all $x\in R$ with $f(x_1,\ldots,x_n)^2$ is central valued in $R$.

Keywords: prime ring, derivation, generalized derivation, $b$-generalized derivation, generalized skew derivation

MSC numbers: 16W25, 16N6