PreprintCC BY 4.0arXiv
Dimensions of a ring and its formal power series ring
Abstract
Understanding the relation between dim R and dim R[[x]] is a classical problem in commutative algebra. For a Noetherian ring R, one has dim R[[x]]=dim R+1, but the general case is considerably more delicate. In 1973, Arnold proved that finite power-series dimension requires the strong finite type (SFT) condition, whereas, in 2002, Coykendall constructed a one-dimensional SFT domain whose power series ring has infinite dimension. The question of Coykendall and Gilmer whether dim R[[x]]<∞ forces dim R[[x]]≤2dim R+1 was answered negatively by Kang and Park in 2009. In this paper, we prove that, as R ranges over the nonzero commutative rings with identity, the finite pairs (dim R,dim R[[x]]) are exactly (0,1) and the pairs (n,m) with 1≤ n<m.
Related
- Finitistic dimensions of formal power series rings
- Substitution groups of formal power series
- A polynomial method over formal power series
- Spherical Completeness, Coherence, and GCD Properties of Formal Power Series and Witt Vector Rings
- The Hilbert Basis Theorem for Non-Associative Skew Power Series Rings and Non-Associative Skew Laurent Series Rings over a Non-Associative Coefficient Ring
BibTeX
@misc{tran2026dimensions,
title = {{Dimensions of a ring and its formal power series ring}},
author = {Viet-Hoang Tran and Thieu N. Vo and Tan M. Nguyen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17854},
eprint = {2609.17854},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17854},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dimensions of a ring and its formal power series ring
AU - Viet-Hoang Tran
AU - Thieu N. Vo
AU - Tan M. Nguyen
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17854
UR - https://arxiv.org/abs/2609.17854
AB - Understanding the relation between dim R and dim R[[x]] is a classical problem in commutative algebra. For a Noetherian ring R, one has dim R[[x]]=dim R+1, but the general case is considerably more delicate. In 1973, Arnold proved that finite power-series dimension requires the strong finite type (SFT) condition, whereas, in 2002, Coykendall constructed a one-dimensional SFT domain whose power series ring has infinite dimension. The question of Coykendall and Gilmer whether dim R[[x]]<∞ forces dim R[[x]]≤2dim R+1 was answered negatively by Kang and Park in 2009. In this paper, we prove that, as R ranges over the nonzero commutative rings with identity, the finite pairs (dim R,dim R[[x]]) are exactly (0,1) and the pairs (n,m) with 1≤ n<m.
ER -