Generalized derivations with nilpotent values on multilinear polynomials.
type: Item
instance of: scholarly article
title: Generalized derivations with nilpotent values on multilinear polynomials.
zbMATH Open document ID: 1114.16033
author: Jer-Shyong Lin
author: Cheng-Kai Liu
published in: Taiwanese Journal of Mathematics
publication date: 2007-01-29T00:00:00Z
review text: Let \(K\) be a commutative ring with identity, \(R\) a \(K\)-algebra, \(f(X)\) a multilinear polynomial over \(K\) in noncommuting variables, and with an invertible coefficient, and \(g\) a generalized derivation of \(R\). The authors show that if \(R\) is prime with no nonzero nil right ideal, \(f(X)\) is not central valued on \(R\), and \(g(f(r_i))\) is nilpotent for all \(r_i\in I\), a nonzero ideal of \(R\), then \(g=0\). When \(R\) is semiprime and when \(g(f(r_i))^n=0\) for all \(r_i\in R\), with \(n\) a fixed positive integer, then \([f(X),y]g(z)\) is an identity for \(R\).
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 16R50
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16U80
zbMATH DE Number: 5119517
zbMATH Keywords: multilinear polynomials
zbMATH Keywords: generalized derivations
reviewed by: Charles Lanski
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500557297
OpenAlex ID: W1624465533
DOI: 10.11650/TWJM/1500557297
description: scientific article; zbMATH DE number 5119517