Krull dimension of mixed extensions

Abstract

For a commutative ring R with identity, dim R shall stand for the Krull dimension of R. It is known that dim R vertical bar x vertical bar <= 2 dim R + 1. We show that this does not hold for the power series extensions. Using mixed extensions, we construct ail example of a finite-dimensional integral domain R such that 2 dim R + 1 < dimR parallel to x parallel to < infinity. Let D be a finite-dimensional SFT prufer domain and D vertical bar x(1)parallel to...vertical bar x(n)parallel to be a mixed extension. According to Arnold, dim D vertical bar x(1).....x(n)vertical bar = dim D + n and dim D parallel to x(1).....x(n)parallel to = n dim D + 1. We generalize Arnold&apos;s result by showing that dim D vertical bar x(1)parallel to...vertical bar x(n)vertical bar = n dim D + 1 provided that there is at least one power series extension. In particular if R = D vertical bar x(1),.... x(n-1)vertical bar and dim D = d > 2(n - 1)/(n - 2), then dim R parallel to x parallel to = dn + 1 > 2 dim R + 1. This is all answer to the question of Coykendall and Gilmer. (C) 2009 Elsevier B.V. All rights reserved