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 $R = \mathbb{Z}[i][[z]] \cap O(\mathbb{D})$ and $R_{\mathbb{R}} = \mathbb{Z}[[z]] \cap O(\mathbb{D})$. We give a trichotomy for maximal ideals, classify the prime ideals lying outside the class $\mathfrak{P}_1$ of primes containing a unit-constant-term element, attach to each element of $\mathfrak{P}_1$ a complete analytic invariant, and construct an injection from ultrafilters on admissible divisors into $\mathfrak{P}_1$ by an ultraproduct device. We prove the point-evaluation ideals $P_a$ are prime for all $a \in \D$ and maximal for real $a$, and deduce a conditional B\'ezout property for disjoint divisors. All results recalled from \cite{PaperI} are used only as cited; no proof of that paper is reproduced here.