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Grant Moles

Publications and source records attributed to Grant Moles.

8 recordsLinked to original sources
arXiv2026

(Locally) Associated Subrings in Polynomial and Power Series Extensions

The associated, ideal-preserving, and locally associated properties of subrings, first formally defined in 2024, give a way of understanding the multiplicative structure of a subring given information about the larger ring. In this paper, we establish notation and preliminary results on a generalized type of polynomial and power series rings that is often used in the construction of counterexamples in the field of commutative algebra. We then provide sets of necessary and sufficient conditions under which such a polynomial or power series ring may be (locally) associated in a larger such ring. This allows for the production of several informative examples, as well as a better understanding of the circumstances under which the ring of formal power series $R[[x]]$ over an order $R$ in a number field may be half-factorial.

math.AC↗
arXiv2026

Elasticity of Orders from the $S$-relative Davenport Constant: an Arithmetic Application of a Number-Theoretic Investigation

Orders in algebraic number fields have long been objects of central interest in algebraic number theory. Despite non-maximal orders failing to be Dedekind, the present authors have previously shown that the structure of the ideal class group may still contain enough information to determine elasticity. In this paper, we develop the $S$-relative Davenport constant, which builds on previous work by M. Ska{\l}ba. Although Ska{\l}ba's original construction was defined to aid in the study of binary quadratic forms, we show that this related invariant is the exact tool needed to tackle the question of elasticity in non-integrally closed orders. In particular, we investigate the elasticity of orders $\mathcal{O}$ whose conductor ideal $I=(\mathcal{O}:\mathcal{O}_K)$ is prime as an ideal of $\mathcal{O}$, as well as orders in quadratic number fields with primary conductor. We also give conditions under which $\mathcal{O}$ will have the same elasticity as the full ring of integers $\mathcal{O}_K$.

math.AC↗
arXiv2025

Locally Associated Orders in Real Quadratic Number Fields

In 2025, the concept of an order in a number field being associated, ideal-preserving, or locally associated was introduced in order to tackle problems in factorization. In this paper, we explore locally associated orders in real quadratic number fields of the form $\mathbb{Q}[\sqrt{p}]$, with $p\in\mathbb{N}$ prime. In particular, we develop strategies and produce results which make determining when a given order in such a number field is (or is not) locally associated much easier. We also highlight the relatively few cases which defy simple characterization, leading to a conjecture on the solutions to Pell's equations of the form $x^2-y^2p=1$.

math.AC↗
arXiv2025

Multiplicative Relationships of Subrings and their Applications to Factorization

When studying the properties of a ring $R$, it is often useful to compare $R$ to other rings whose properties are already known. In this paper, we define three ways in which a subring $R$ might be compared to a larger ring $T$: being associated, being ideal-preserving, or being locally associated. We then explore how these properties of a subring might be leveraged to give information about $R$, including applications to the field of factorization. Of particular interest is the result that an order in a number field is associated if and only if it is both ideal-preserving and locally associated. We conclude with a discussion of how these properties are realized in the case of orders in a number field and how such orders might be found.

math.AC↗
arXiv2025

Elasticity in orders of an algebraic number field with radical conductor ideal and their rings of formal power series

Orders in an algebraic number field form a class of rings which are of special historical interest to the field of factorization theory. One of the primary tools used to study factorization is elasticity - a measure of how badly unique factorization fails in a domain. This paper explores properties of orders in a number field and how they can be used to study elasticity in not only the orders themselves, but also in rings of formal power series over the orders. Of particular interest is the fact, proven here, that power series extensions in finitely many variables over half-factorial rings of algebraic integers must themselves be half-factorial. It is also shown that the HFD property is not preserved in general for power series rings over non-integrally closed orders in a number field.

math.AC↗
arXiv2025

Elasticity of Orders with Prime Conductor

Let $R$ be an order in a number field whose conductor ideal $P := (R:\overline{R})$ is prime in the ring of integers $\overline{R}$. In this paper, we explore the factorization properties of such orders. Most notably, we give a complete characterization of the elasticity of $R$ in terms of its class group. We conclude with an application to the computation of class groups of certain orders.

math.AC↗
arXiv2025

Overrings of half-factorial orders

The behavior of factorization properties in various ring extensions is a central theme in commutative algebra. Classically, the UFDs are (completely) integrally closed and tend to behave well in standard ring extensions, with the notable exception of power series extension. The half-factorial property is not as robust; HFDs need not be integrally closed and the half-factorial property is not necessarily preserved in integral extensions or even localizations. Here we exhibit classes of HFDs that behave well in (almost) integral extensions, resolve an open question on the behavior of the boundary map, and give a squeeze theorem for elasticity in certain domains.

math.AC↗