Density of Polynomial Maps

Canadian mathematical bulletin

Chen-Lian Chuang¹  Tsiu-Kwen Lee¹ 

[1] Department of Mathematics, National Taiwan University, Taipei 106, Taiwan

关键词: density;    polynomial;    endomorphism ring;    PI;   

DOI  :  10.4153/CMB-2010-041-1

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

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 $0e ain R$. If $af(x_1,dots,x_m)^{n(x_i)}=0$ for all $x_1,dots,x_min R$, 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.

【 授权许可】

Unknown