Arithmetic of semisubtractive semidomains

Abstract

A subset [Formula: see text] of an integral domain is called a semidomain if the pairs [Formula: see text] and [Formula: see text] are commutative and cancellative semigroups with identities. The multiplication of [Formula: see text] extends to the group of differences [Formula: see text], turning [Formula: see text] into an integral domain. In this paper, we study the arithmetic of semisubtractive semidomains (i.e. semidomains [Formula: see text] for which either [Formula: see text] or [Formula: see text] for every [Formula: see text]). 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 paper, we present examples to provide insight into the arithmetic aspects of semisubtractive semidomains.