CMS/SMC
Canadian Mathematical Society
www.cms.math.ca
Canadian Mathematical Society
  location:  PublicationsjournalsCMB
Publications        
Abstract view

Density of Polynomial Maps

  Published:2010-04-06
 Printed: Jun 2010
  • Chen-Lian Chuang,
    Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
  • Tsiu-Kwen Lee,
    Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
Features coming soon:
Citations   (via CrossRef) Tools: Search Google Scholar:
Format:   HTML   LaTeX   MathJax  

Abstract

Let $R$ be a dense subring of $\operatorname{End}(_DV)$, where $V$ is a left vector space over a division ring $D$. If $\dim{_DV}=\infty$, then the range of any nonzero polynomial $f(X_1,\dots,X_m)$ on $R$ is dense in $\operatorname{End}(_DV)$. As an application, let $R$ be a prime ring without nonzero nil one-sided ideals and $0\ne a\in R$. If $af(x_1,\dots,x_m)^{n(x_i)}=0$ for all $x_1,\dots,x_m\in R$, where $n(x_i)$ is a positive integer depending on $x_1,\dots,x_m$, then $f(X_1,\dots,X_m)$ is a polynomial identity of $R$ unless $R$ is a finite matrix ring over a finite field.
Keywords: density, polynomial, endomorphism ring, PI density, polynomial, endomorphism ring, PI
MSC Classifications: 16D60, 16S50 show english descriptions Simple and semisimple modules, primitive rings and ideals
Endomorphism rings; matrix rings [See also 15-XX]
16D60 - Simple and semisimple modules, primitive rings and ideals
16S50 - Endomorphism rings; matrix rings [See also 15-XX]
 

© Canadian Mathematical Society, 2014 : http://www.cms.math.ca/