Overrings of half-factorial orders
Jason Boynton, Jim Coykendall, Grant Moles, Chelsey Morrow
TL;DR
We address the stability of half-factorial factorization in overrings of orders in rings of algebraic integers, focusing on overrings and integral extensions of an $R$ that is an HFD. Using the boundary map $\partial_R$ and radical conductor assumptions, we derive structural results showing that the integral closure often preserves the HFD property, and establish a Squeeze-Theorem: if the conductor $I=(\overline{R}:R)$ is radical, then every intermediate domain is HFD, with several equivalent formulations tying irreducibles to boundary values. The findings offer affirmative resolutions to questions about boundary behavior and extend the HFD stability picture to overrings beyond rings of integers, providing tools for elasticity control in 1-dimensional Noetherian settings.
Abstract
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.