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1. CMB 2010 (vol 53 pp. 223)
Density of Polynomial Maps 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 Categories:16D60, 16S50 |
2. CMB 2008 (vol 51 pp. 460)
On Primitive Ideals in Graded Rings Let $R=\bigoplus_{i=1}^{\infty}R_{i}$ be a graded nil ring. It is shown
that primitive ideals in $R$ are homogeneous. Let
$A=\bigoplus_{i=1}^{\infty}A_{i}$ be a graded non-PI just-infinite
dimensional algebra and let $I$ be a prime ideal in $A$. It is shown
that either $I=\{0\}$ or $I=A$. Moreover, $A$ is either primitive or
Jacobson radical.
Categories:16D60, 16W50 |
3. CMB 2005 (vol 48 pp. 587)
Separation of Variables for $U_{q}(\mathfrak{sl}_{n+1})^{+}$ Let $U_{q}(\SL)^{+}$ be the positive part of the quantized enveloping
algebra $U_{q}(\SL)$. Using results of Alev--Dumas and Caldero related
to the center of $U_{q}(\SL)^{+}$, we show that this algebra is free
over its center. This is reminiscent of Kostant's separation of
variables for the enveloping algebra $U(\g)$ of a complex semisimple
Lie algebra $\g$, and also of an analogous result of Joseph--Letzter
for the quantum algebra $\Check{U}_{q}(\g)$. Of greater importance to
its representation theory is the fact that $\U{+}$ is free over a
larger polynomial subalgebra $N$ in $n$ variables. Induction from $N$
to $\U{+}$ provides infinite-dimensional modules with good properties,
including a grading that is inherited by submodules.
Categories:17B37, 16W35, 17B10, 16D60 |