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# 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
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## 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
 MSC Classifications: 16D60 - Simple and semisimple modules, primitive rings and ideals 16S50 - Endomorphism rings; matrix rings [See also 15-XX]