标题： 唯一因子分解半域的局部化 显示英文标题

标题： Localization of unique factorization semidomains

摘要： 半环域是一个整环的子半环。 在这一类中，唯一因子分解半环域（UFS）的特征是每个非零、非单位元素都可以分解为有限多个素元素的乘积。 在本文中，我们研究半环域的局部化，特别关注UFS。 我们证明UFS的局部化仍然是UFS，从而得出结论，即UFS要么是一个唯一因子分解域，要么是加法约化的。 此外，我们提供了一个子半环$\mathfrak{S}$的例子，它是$\mathbb{R}$的子半环，使得$(\mathfrak{S}, \cdot)$和$(\mathfrak{S}, +)$都是半因子的，这有助于解决Baeth、Chapman和Gotti提出的猜想。 显示英文摘要

摘要： A semidomain is a subsemiring of an integral domain. Within this class, a unique factorization semidomain (UFS) is characterized by the property that every nonzero, nonunit element can be factored into a product of finitely many prime elements. In this paper, we investigate the localization of semidomains, focusing specifically on UFSs. We demonstrate that the localization of a UFS remains a UFS, leading to the conclusion that a UFS is either a unique factorization domain or is additively reduced. In addition, we provide an example of a subsemiring $\mathfrak{S}$ of $\mathbb{R}$ such that $(\mathfrak{S}, \cdot)$ and $(\mathfrak{S}, +)$ are both half-factorial, shedding light on a conjecture posed by Baeth, Chapman, and Gotti.