Factorizations in algebraic monogenic semidomains
Abstract
We study the additive and multiplicative arithmetic of algebraic monogenic semidomains of the form Sα. We first relate their structure to the conjugates of α, characterize when Sα is a ring, and determine when its additive monoid is free. We then investigate units and prove a Dirichlet-type theorem showing that Sα× 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 α is positive. We also give a criterion that produces non-atomic algebraic monogenic semidomains. Turning to finiteness properties, we prove that ℤ[α] is a finite factorization domain for every algebraic number α, establish criteria ensuring that Sα 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 ℤ[β] for arbitrary algebraic β, and prove that, for algebraic integers α, unique factorization ascends from ℤ[α] to ℤ[α/n] for every n∈ℕ.
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BibTeX
@misc{dani2026factorizations,
title = {{Factorizations in algebraic monogenic semidomains}},
author = {Jiya Dani and Felix Gotti and Bryan Li and Arav Paladiya},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32138},
eprint = {2609.32138},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32138},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Factorizations in algebraic monogenic semidomains
AU - Jiya Dani
AU - Felix Gotti
AU - Bryan Li
AU - Arav Paladiya
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32138
UR - https://arxiv.org/abs/2609.32138
AB - We study the additive and multiplicative arithmetic of algebraic monogenic semidomains of the form S_α. We first relate their structure to the conjugates of α, characterize when S_α is a ring, and determine when its additive monoid is free. We then investigate units and prove a Dirichlet-type theorem showing that S_α^× 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 α is positive. We also give a criterion that produces non-atomic algebraic monogenic semidomains. Turning to finiteness properties, we prove that ℤ[α] is a finite factorization domain for every algebraic number α, establish criteria ensuring that S_α 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 ℤ[β] for arbitrary algebraic β, and prove that, for algebraic integers α, unique factorization ascends from ℤ[α] to ℤ[α/n] for every n∈ℕ.
ER -