有理单基因半域中的因式分解
Factorizations in rational monogenic semidomains
浏览论文内容
中文总结 AI 辅助
研究由有理参数\(q\)生成的单基因半域\(S_q\)的因式分解，引入技术分数幺半群\(T_q\)，研究其多种因式分解性质，包括UF、FF、BF等，确定相关参数值及性质等价关系，证明\(S_q\)是克鲁尔半域的条件。
中文摘要 AI 辅助
对于\(\alpha\in\mathbb{C}\)，由\(\alpha\)生成的单基因半域是复域\(\mathbb{C}\)中包含\(\alpha\)的最小子半环\(S_\alpha\)。我们对由有理参数\(q\)生成的单基因半域\(S_q\)的算术和因式分解进行系统研究。引入并研究技术分数的幺半群\(T_q\)，它是\(S_q\)乘法幺半群的一个除数封闭子幺半群，编码了\(S_q\)的大量算术信息。接着研究\(S_q\)的几个基本因式分解性质，如有界因式分解（BF）、有限因式分解（FF）、唯一因式分解（UF）和半因式分解（HF）性质。证明了\(S_q\)满足UF性质当且仅当它满足HF性质，确定了\(S_q\)满足FF性质的所有正有理参数值，表明在有理单基因半域类中，BF性质等价于主理想的升链条件，最后证明\(S_q\)是克鲁尔半域当且仅当它是根封闭的，即恰好当\(S_q\)满足UF性质时。
英文摘要
For $α\in \mathbb{C}$, the monogenic semidomain generated by $α$ is the smallest subsemiring $S_α$ of the complex field $\mathbb{C}$ containing $α$. We initiate a systematic study of the arithmetic and factorizations of the monogenic semidomains $S_q$ generated by rational parameters $q$. After some preliminaries, we introduce and investigate the monoid of technical fractions $T_q$, which is a divisor-closed submonoid of the multiplicative monoid of $S_q$ that encodes a significant amount of arithmetic information about $S_q$. We then study several fundamental factorization properties of $S_q$: the bounded factorization (BF) and finite factorization (FF) properties, the unique factorization (UF) property, and the half-factorial (HF) property. First, we prove that $S_q$ satisfies the UF property if and only if it satisfies the HF property, which happens when $q \in \mathbb{N} \cup \mathbb{N}^{-1}$. We determine all the positive rational values of the parameter $q$ for which $S_q$ satisfies the FF property. Then we show that, over the class of rational monogenic semidomains, the BF property is equivalent to the ascending chain condition on principal ideals. Finally, we prove that $S_q$ is a Krull semidomain if and only if it is root-closed, which happens precisely when $S_q$ satisfies the UF property.
