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 ¥?

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Last modified: April 17, 2002