Elasticity in orders of an algebraic number field with radical conductor ideal and their rings of formal power series
James Barker Coykendall, Grant Moles
TL;DR
The work investigates how elasticity and half-factoriality transfer between orders in number fields and their rings of formal power series. By developing the notions of associated, ideal-preserving, and locally associated subrings, it establishes inequalities and equalities between the elasticities of $R$ and its integral closure $ar{R}$, and extends these results to $R[[x]]$ relative to $ar{R}[[x]]$. A key outcome is that, whenever the conductor ideal is radical, $R$ is an $HFD$ if and only if $R[[x]]$ is an $HFD$, with the equality of elasticities $ ho(R[[x]])= ho(ar{R}[[x]])$ mirroring $ ho(R)= ho(ar{R})$; this yields concrete corollaries for quadratic orders. The paper also shows that non-radical conductor ideals can disrupt these preservation properties, underscoring the necessity of radicalness in the main theorems. Overall, the results deepen factorization theory for orders and their power-series extensions in algebraic number fields.
Abstract
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.