Power series over integral domains of Krull type

Abstract

An integral domain [Formula: see text] is said to be of Krull type if [Formula: see text] is a locally finite intersection of essential valuation overrings [Formula: see text] of [Formula: see text]. If each [Formula: see text] is required to be one-dimensional and discrete, then [Formula: see text] is called a Krull domain. In this paper, we show that if [Formula: see text] is an integral domain of Krull type such that some [Formula: see text] is not an SFT ring, then the power series ring [Formula: see text] is not a locally finite intersection of valuation domains. This is a generalization of our previous work, where [Formula: see text] is assumed to be a valuation domain. It follows that [Formula: see text] is a Krull domain if and only if both [Formula: see text] and [Formula: see text] are integral domains of Krull type, which is an improvement of a result by Paran and Temkin. We also prove that if [Formula: see text] is a Prüfer domain, then [Formula: see text] is a Krull domain if and only if [Formula: see text] is an integral domain of Krull type.