Subrings of the power series ring over a principal ideal domain

Abstract

Let D be a principal ideal domain (PID), I be an ideal of D, and X be an indeterminate over D. Let [D;I][X] be the subring of the power series ring $D[\mathrm{}[X]\mathrm{}]$ consisting of all power series $f={\sum }_{i=0}^{\infty }{a}_i{X}^i$ in $D[\mathrm{}[X]\mathrm{}]$ such that $a_i\in I$ for all large i. By definition, the polynomial ring $D[X]$ and the power series ring $D[\mathrm{}[X]\mathrm{}]$ are special cases of $[D;I][X]$ when $I=(0)$ and I = D, respectively. In this article, we investigate the ring $R:=[D;I][X]$ in the case I is a nonzero proper ideal of D. We prove that R is a two-dimensional non-Noetherian ring. For each maximal ideal P of D, it is shown that $P[\mathrm{}[X]\mathrm{}]\cap R=\sqrt{PR}$ is a height-one prime ideal of R. The set of units of R is given and the spectrum of R is also described. Unlike the power series ring $D[\mathrm{}[X]\mathrm{}],$ the ring R is not a unique factorization domain (UFD). Furthermore, when I is a nonzero prime ideal, R does not satisfy both ACCP and the atomic property. In obtaining results on R, we introduce and sometimes use results on the ring R_S, where I = dD with $0\cancel{=}d\in D$ and $S=\left\{d^n\right|n\ge 0\}.$ Closely related to R, the ring R_S is shown to be a Noetherian UFD with Krull dimension at most two. Moreover, R_S has Krull dimension two exactly when I is not contained in the Jacobson radical of D; otherwise R_S is a PID.

Keywords:

• Krull dimension
• PID
• power series ring
• prime ideal
• UFD

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13A15
• 13C15
• 13F10: 13F15
• 13F25

Acknowledgment

The authors would like to thank the referee for his/her remarks, which help us improve the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first-named author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B06029867). The second-named author is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.04-2019.06.