圆盘上整系数幂级数环的素谱
The prime spectrum of rings of integer-coefficient power series on the disk
AI总结：
研究圆盘上整系数幂级数环的素谱，给出极大理想三分法、素理想分类及超滤子注入，证明点赋值理想素性与条件Bézout性质。
AI中文摘要：
这是对\cite{PaperI}的后续研究，其中证明了开单位圆盘$\D$上的离散有效因子是某个具有整数泰勒系数的全纯函数的零因子，当且仅当该因子在复共轭下不变。基于该实现定理以及其中建立的单位刻画，我们研究了环$R = \mathbb{Z}[i][[z]] \cap O(\mathbb{D})$和$R_{\mathbb{R}} = \mathbb{Z}[[z]] \cap O(\mathbb{D})$的素谱与极大谱。我们给出了极大理想的三分法，对位于包含单位常数项元素的素理想类$\mathfrak{P}_1$之外的素理想进行了分类，为$\mathfrak{P}_1$中的每个元素赋予一个完全解析不变量，并通过超积构造建立了从可容许因子上的超滤子到$\mathfrak{P}_1$的单射。我们证明了点赋值理想$P_a$对所有$a \in \D$都是素理想，对实数$a$是极大理想，并推导了不相交因子的条件Bézout性质。所有从\cite{PaperI}引用的结果仅按引用使用；此处不重现该论文的任何证明。
英文摘要：
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 Bézout property for disjoint divisors. All results recalled from \cite{PaperI} are used only as cited; no proof of that paper is reproduced here.
