发表机构
MIT; Yale University; UC Berkeley（麻省理工学院; 耶鲁大学; 加州大学伯克利分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究代数幺半生成半整环的算术性质，刻画其结构、单位群和有限因式分解性，并给出有理元素隶属判据及唯一因式分解的上升定理。
AI 中文摘要
我们研究形如 $S_\alpha$ 的代数幺半生成半整环的加法和乘法算术性质。首先，我们将其结构与 $\alpha$ 的共轭联系起来，刻画 $S_\alpha$ 何时是环，并确定其加法幺半群何时是自由的。接着，我们研究单位并证明一个 Dirichlet 型定理，表明 $S_\alpha^\times$ 是有限循环单位根群与有限生成自由阿贝尔群的乘积；特别地，当 $\alpha$ 为正时，单位群是自由阿贝尔的。我们还给出一个产生非原子代数幺半生成半整环的判据。转向有限性性质，我们证明对于每个代数数 $\alpha$，$\mathbb{Z}[\alpha]$ 是有限因式分解整环，建立确保 $S_\alpha$ 是有限因式分解半整环的判据，并构造无穷多个正三次生成元，产生非环的有限因式分解半整环。最后，我们给出任意代数数 $\beta$ 的 $\mathbb{Z}[\beta]$ 中有理元素的隶属判据，并证明对于代数整数 $\alpha$，唯一因式分解从 $\mathbb{Z}[\alpha]$ 上升到 $\mathbb{Z}[\alpha/n]$ 对每个 $n\in\mathbb{N}$ 成立。
英文摘要
We study the additive and multiplicative arithmetic of algebraic monogenic semidomains of the form $S_α$. We first relate their structure to the conjugates of $α$, characterize when $S_α$ is a ring, and determine when its additive monoid is free. We then investigate units and prove a Dirichlet-type theorem showing that $S_α^\times$ is the product of a finite cyclic group of roots of unity and a finitely generated free abelian group; in particular, the unit group is free abelian when $α$ is positive. We also give a criterion that produces non-atomic algebraic monogenic semidomains. Turning to finiteness properties, we prove that $\mathbb{Z}[α]$ is a finite factorization domain for every algebraic number $α$, establish criteria ensuring that $S_α$ is a finite factorization semidomain, and construct infinitely many positive cubic generators yielding finite factorization semidomains that are not rings. Finally, we give a membership criterion for rational elements of $\mathbb{Z}[β]$ for arbitrary algebraic $β$, and prove that, for algebraic integers $α$, unique factorization ascends from $\mathbb{Z}[α]$ to $\mathbb{Z}[α/n]$ for every $n\in\mathbb{N}$.
Comments28 pages
