Density of Polynomial Maps

Chuang, Chen-Lian ; Lee, Tsiu-Kwen

Canadian mathematical bulletin, Tome 53 (2010) no. 2, pp. 223-229

Résumé

Let $R$ be a dense subring of $\text{End}\left( _{D}V \right)$ , where $V$ is a left vector space over a division ring $D$ . If $\dim{{\,}_{D}}V\,=\,\infty $ , then the range of any nonzero polynomial $f\left( {{X}_{1}},\,\ldots \,,\,{{X}_{m}} \right)$ on $R$ is dense in $\text{End}\left( _{D}V \right)$ . As an application, let $R$ be a prime ring without nonzero nil one-sided ideals and $0\,\ne \,a\,\in \,R$ . If $af{{\left( {{x}_{1}},\ldots ,{{x}_{m}} \right)}^{n\left( {{x}_{i}} \right)}}\,=\,0$ for all ${{x}_{1}},\,\ldots \,,\,{{x}_{m}}\,\in \,R$ , where $n\left( {{x}_{i}} \right)$ is a positive integer depending on ${{x}_{1}},\,\ldots \,,\,{{x}_{m}}\,\in \,R$ , then $f\left( {{X}_{1}},\,\ldots \,,\,{{X}_{m}} \right)$ is a polynomial identity of $R$ unless $R$ is a finite matrix ring over a finite field.

Détail

DOI : 10.4153/CMB-2010-041-1

Mots-clés : 16D60, 16S50, density, polynomial, endomorphism ring, PI