Krull Dimension in Power Series Ring Over an Almost Pseudo-Valuation Domain

Abstract

Let R be an integral domain. We say that R is a star-domain if R has at least a height one prime ideal and if for each height one prime ideal P of R, R satisfies the acc on P-principal ideals (i.e., ideals of the form aP, a ∈ R). We prove that if R is an APVD with nonzero finite Krull dimension, then the power series ring R[[X]] has finite Krull dimension if and only if R is a residually star-domain (i.e., for each nonmaximal prime ideal P of R, R/P is a star-domain) if and only if R[[X]] is catenarian.

Key Words:

• APVD
• Krull dimension
• Power series ring

2000 Mathematics Subject Classification:

• 13A15
• 13A18
• 13C15
• 13F05
• 13F25

ACKNOWLEDGMENT

The authors thank the referee for helpful suggestions concerning presentation. The first author express his sincere gratitude to the second author for his guidance and encouragement. Many thanks go also to Professors M. Fontana, D. E. Dobbs, A. Badawi and M. H. Park for their helpful enlightenment.

Notes

Communicated by I. Swanson.