FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

V. A. Mushrub

Abstract

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```
Let
```$R$ be an associative ring,
$X = \{x_i\colon\ i \in \Gamma\}$
be a nonempty set
of variables, $F =
\{f_i\colon\ i \in \Gamma\}$ 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 $Aa \ne 0$
for all nonzero $a \in A$ .
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
$\dim R < \infty$ ?

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Last modified: April 17, 2002