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 = {xi: i Î G} be a nonempty set of variables, F = {fi: 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 < ¥?

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