FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2001, VOLUME 7, NUMBER 4, PAGES 1107-1121

**On the uniform dimension of skew polynomial rings in many variables**

V. A. Mushrub

Abstract

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Let $R$ be
an associative ring, $X\; =\; \{x$_{i}: i Î
G} be
a nonempty set of variables, $F\; =\; \{f$_{i}: i Î
G} be a family
of injective ring endomorphisms of $R$ and $A(R,F)$ be the Cohn--Jordan
extension.
In this paper we prove that the left uniform dimension of
the skew polynomial ring $R[X,F]$ is equal to
the left uniform dimension of $A(R,F)$, provided that
$Aa$¹
0 for all nonzero $a$Î
A.
Furthermore, we show that for semiprime rings the equality
$dim\; R\; =\; dim\; R[X,F]$
does not hold in the general case.
The following problem is still open.
Does $dim\; R\; =\; dim\; R[x,f]$ hold if
$R$ is
a semiprime ring, $f$ is an injective ring
endomorphism of $R$ and $dim\; R\; <$¥?

All articles are
published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/k01/k014/k01409h.htm

Last modified: April 17, 2002