The Krull dimension of power series rings over non-SFT rings

Abstract

Let R be a commutative ring with identity. We show that the Krull dimension of the power series ring R[[X]] can be uncountably infinite, i.e., there exists an uncountably infinite chain of prime ideals in R[[X]], even if dim R is finite. In fact, we show that dim R[[X]] is uncountably infinite if R is a non-SFT ring, which is an improvement of Arnold's result. (C) 2012 Elsevier B.V. All rights reserved.