Factorizations in algebraic monogenic semidomains

Abstract

We study the additive and multiplicative arithmetic of algebraic monogenic semidomains of the form $S_{\alpha }$. We first relate their structure to the conjugates of $\alpha$, characterize when $S_{\alpha }$ is a ring, and determine when its additive monoid is free. We then investigate units and prove a Dirichlet-type theorem showing that $S_{\alpha }^{\times }$ is the product of a finite cyclic group of roots of unity and a finitely generated free abelian group; in particular, the unit group is free abelian when $\alpha$ is positive. We also give a criterion that produces non-atomic algebraic monogenic semidomains. Turning to finiteness properties, we prove that $\mathbb{Z}[\alpha ]$ is a finite factorization domain for every algebraic number $\alpha$, establish criteria ensuring that $S_{\alpha }$ is a finite factorization semidomain, and construct infinitely many positive cubic generators yielding finite factorization semidomains that are not rings. Finally, we give a membership criterion for rational elements of $\mathbb{Z}[\beta ]$ for arbitrary algebraic $\beta$, and prove that, for algebraic integers $\alpha$, unique factorization ascends from $\mathbb{Z}[\alpha ]$ to $\mathbb{Z}[\alpha /n]$ for every $n\in \mathbb{N}$.

arXiv comment28 pages