Almost strong finite type rings and Krull dimension of power series ring extensions from sequences

Abstract

Let [Formula: see text] be the collection of power series with coefficients in a commutative ring [Formula: see text] with identity. For a suitable function [Formula: see text] from [Formula: see text] to [Formula: see text], one can define a multiplication [Formula: see text] in [Formula: see text] such that together with the usual addition, [Formula: see text] becomes a ring that contains [Formula: see text] as a subring. Denote this ring by [Formula: see text]. By this observation, the usual power series ring and the well-known Hurwitz series ring are the special cases of [Formula: see text] when [Formula: see text] for all [Formula: see text] and [Formula: see text] for all [Formula: see text], respectively. In this paper, we study the Krull dimension of [Formula: see text]. We first introduce the concept of almost strong finite type (ASFT) rings and study basic properties of this type of rings. We then show that for any function [Formula: see text], the Krull dimension of [Formula: see text] is at least [Formula: see text] if [Formula: see text] is a non-ASFT ring, which is an analogue of the result about the Krull dimension of the usual power series ring that [Formula: see text] if [Formula: see text] is a non-SFT ring. In particular, we have the Krull dimension of the Hurwitz series ring is at least [Formula: see text] if [Formula: see text] is a non-ASFT ring, which gives an answer to a question of Benhissi and Koja about the infiniteness of the Krull dimension of the Hurwitz series ring.