The prime spectrum of rings of integer-coefficient power series on the disk
Abstract
This is a sequel to \cite{PaperI}, where it is shown that a discrete effective divisor on the open unit disk D is the zero divisor of a holomorphic function with integer Taylor coefficients if and only if it is invariant under complex conjugation. Building on that realization theorem and on the unit characterization established there, we study the prime and maximal spectra of the rings R = ℤ[i][[z]] ∩ O(𝔻) and Rℝ = ℤ[[z]] ∩ O(𝔻). We give a trichotomy for maximal ideals, classify the prime ideals lying outside the class P1 of primes containing a unit-constant-term element, attach to each element of P1 a complete analytic invariant, and construct an injection from ultrafilters on admissible divisors into P1 by an ultraproduct device. We prove the point-evaluation ideals Pa are prime for all a ∈ D and maximal for real a, and deduce a conditional Bézout property for disjoint divisors. All results recalled from \cite{PaperI} are used only as cited; no proof of that paper is reproduced here.
Related
- The Hilbert Basis Theorem for Non-Associative Skew Power Series Rings and Non-Associative Skew Laurent Series Rings over a Non-Associative Coefficient Ring
- On the Krull dimensions of rings of integer-valued polynomials and polynomial rings
- Stacked-bases Properties, maximal ideals and polynomial closures of Rings of Integer-Valued Polynomials
- Integer-valued polynomials over structural matrix rings
- Prime Plus a Non-Square-Free Integer: An Elementary Approach
BibTeX
@misc{bannon2026prime,
title = {{The prime spectrum of rings of integer-coefficient power series on the disk}},
author = {Jon Bannon and David Feldman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26812},
eprint = {2609.26812},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26812},
}
RIS
TY - UNPB
TI - The prime spectrum of rings of integer-coefficient power series on the disk
AU - Jon Bannon
AU - David Feldman
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26812
UR - https://arxiv.org/abs/2609.26812
AB - This is a sequel to \cite{PaperI}, where it is shown that a discrete effective divisor on the open unit disk D is the zero divisor of a holomorphic function with integer Taylor coefficients if and only if it is invariant under complex conjugation. Building on that realization theorem and on the unit characterization established there, we study the prime and maximal spectra of the rings R = ℤ[i][[z]] ∩ O(𝔻) and R_ℝ = ℤ[[z]] ∩ O(𝔻). We give a trichotomy for maximal ideals, classify the prime ideals lying outside the class P_1 of primes containing a unit-constant-term element, attach to each element of P_1 a complete analytic invariant, and construct an injection from ultrafilters on admissible divisors into P_1 by an ultraproduct device. We prove the point-evaluation ideals P_a are prime for all a ∈ D and maximal for real a, and deduce a conditional Bézout property for disjoint divisors. All results recalled from \cite{PaperI} are used only as cited; no proof of that paper is reproduced here.
ER -