半减半域算术

Pub Date : 2023-11-13 DOI:10.1142/s0219498825501634
Hannah Fox, Agastya Goel, Sophia Liao
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引用次数: 0

摘要

如果$(S,+)$和$(S\setminus\{0\}, \cdot)$这两对半群是交换半群和可取消半群,那么一个积分域的子集$S$被称为半域。$S$ 的乘法扩展到差集 $\mathscr{G}(S)$,把 $\mathscr{G}(S)$ 变成了一个积分域。在本文中,我们将研究半减半域的算术(即对于每一个 $s \in S$ 或 $-s \in S$ 的半域 $S$)。具体来说,我们提供了半减法半域成为原子半域、满足主ideals的升链条件、成为有界因式分解半域以及成为有限因式分解半域的必要条件和充分条件,这些条件是对具有唯一因式分解这一性质的后续放宽。此外,我们还介绍了因式半域和半因式半域的特征。在整篇文章中,我们举例说明了半减半域的算术方面。
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Arithmetic of Semisubtractive Semidomains
A subset $S$ of an integral domain is called a semidomain if the pairs $(S,+)$ and $(S\setminus\{0\}, \cdot)$ are commutative and cancellative semigroups with identities. The multiplication of $S$ extends to the group of differences $\mathscr{G}(S)$, turning $\mathscr{G}(S)$ into an integral domain. In this paper, we study the arithmetic of semisubtractive semidomains (i.e., semidomains $S$ for which either $s \in S$ or $-s \in S$ for every $s \in \mathscr{G}(S)$). Specifically, we provide necessary and sufficient conditions for a semisubtractive semidomain to be atomic, to satisfy the ascending chain condition on principals ideals, to be a bounded factorization semidomain, and to be a finite factorization semidomain, which are subsequent relaxations of the property of having unique factorizations. In addition, we present a characterization of factorial and half-factorial semisubtractive semidomains. Throughout the article, we present examples to provide insight into the arithmetic aspects of semisubtractive semidomains.
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