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

Let R[[X]] be the collection of power series with coefficients in a commutative ring R with identity. For a suitable function ? from N0 to N, one can define a multiplication *? in R[[X]] such that together with the usual addition, R[[X]] becomes a ring that contains R as a subring. Denote this ring by R-?[[X]]. By this observation, the usual power series ring and the well-known Hurwitz series ring are the special cases of R-?[[X]] when ?(i) = 1 for all i and ?(i) = i! for all i, respectively. In this paper, we study the Krull dimension of R-?[[X]]. 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 ?, the Krull dimension of R-?[[X]] is at least 2 aleph 1 if R is a non-ASFT ring, which is an analogue of the result about the Krull dimension of the usual power series ring that dim R[[X]] >= 2 aleph 1 if R is a non-SFT ring. In particular, we have the Krull dimension of the Hurwitz series ring is at least 2 aleph 1 if R 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.