Arithmetic of semisubtractive semidomains

Abstract

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 $\mathcal{G}(S)$, turning $\mathcal{G}(S)$ 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 $s\in \mathcal{G}(S)$). 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.

15 pages