Topological aspects of the ideal theory in rings of integer-valued polynomials

Finocchiaro, Carmelo; Loper, K.

doi:10.1090/proc/16740

Topological aspects of the ideal theory in rings of integer-valued polynomials
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by Carmelo Antonio Finocchiaro and K. Alan Loper;

Proc. Amer. Math. Soc. 152 (2024), 1809-1819

DOI: https://doi.org/10.1090/proc/16740

Published electronically: March 26, 2024

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Abstract:

Let $D$ be a Dedekind domain with finite residue fields. We provide topological insights into certain classes of ideals of $Int(D)$ lying over a given maximal ideal $\mathfrak m$ of $D$. We completely determine invertible/divisorial ideals in terms of topological properties of subsets of the $\mathfrak m$-adic completion of $D$. Moreover, these results are naturally extended to overrings of $Int(D)$. As an application we provide explicit constructions of divisorial ideals of $Int(D)$ which are not finitely generated.

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