On the arithmetic of polynomial semidomains

TL;DR

This paper explores the arithmetic and divisibility properties of semidomains, a generalization of integral domains, focusing on how these properties ascend to polynomial semidomains and providing examples to illustrate the concepts.

Contribution

It extends the study of factorization properties from integral domains to semidomains, analyzing their ascent to polynomial semidomains and providing new insights and examples.

Findings

ASC property ascends to polynomial semidomains

Bounded and finite factorizations ascend in semidomains

Atomicity and unique factorization do not necessarily ascend

Abstract

A subset $S$ of an integral domain $R$ is called a semidomain provided that the pairs $(S,+)$ and $(S, \cdot)$ are semigroups with identities. The study of factorizations in integral domains was initiated by Anderson, Anderson, and Zafrullah in 1990, and this area has been systematically investigated since then. In this paper, we study the divisibility and arithmetic of factorizations in the more general context of semidomains. We are specially concerned with the ascent of the most standard divisibility and factorization properties from a semidomain to its semidomain of (Laurent) polynomials. As in the case of integral domains, here we prove that the properties of satisfying the ascending chain condition on principal ideals, having bounded factorizations, and having finite factorizations ascend in the class of semidomains. We also consider the ascent of the property of being atomic and that of having unique factorization (none of them ascends in general). Throughout the paper we provide several examples aiming to shed some light upon the arithmetic of factorizations of semidomains.