Localization of unique factorization semidomains

TL;DR

This paper studies the localization properties of unique factorization semidomains (UFSs), showing they remain UFSs after localization and exploring their structural characteristics and examples related to half-factoriality.

Contribution

It proves that localizing a UFS preserves its unique factorization property and provides an example of a subsemiring with both multiplicative and additive half-factoriality.

Findings

Localization of a UFS remains a UFS.

A UFS is either a UFD or additively reduced.

Constructed example of a subsemiring with half-factoriality in both operations.

Abstract

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.