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

TL;DR

This paper investigates the elasticity of orders in algebraic number fields with radical conductor ideals and their formal power series rings, revealing conditions under which half-factoriality is preserved or lost.

Contribution

It demonstrates that power series over half-factorial rings of algebraic integers remain half-factorial, but this property may not hold for non-integrally closed orders.

Findings

Power series over half-factorial algebraic integer rings are half-factorial.

HFD property is not preserved in power series over non-integrally closed orders.

Elasticity properties are linked to the structure of orders and their extensions.

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.