Radical factorization in higher dimension

TL;DR

This paper extends radical factorization theory to strongly discrete Pr"ufer domains, characterizing when finitely generated ideals have radical factorizations and analyzing the structure of the invertible ideals group.

Contribution

It generalizes radical factorization from Dedekind to Pr"ufer domains and characterizes conditions for radical factorizations based on maximal ideals.

Findings

Radical factorization characterized for finitely generated ideals in strongly discrete Pr"ufer domains.

Invertible ideals group often free, especially when spectrum is Noetherian or in rings of integer-valued polynomials.

Provides criteria involving critical maximal ideals for radical factorizations.

Abstract

We generalize the theory of radical factorization from almost Dedekind domain to strongly discrete Pr\"ufer domains; we show that, for a fixed subset $X$ of maximal ideals, the finitely generated ideals with $\mathcal{V}(I)\subseteq X$ have radical factorization if and only if $X$ contains no critical maximal ideals with respect to $X$. We use these notions to prove that in the group $\mathrm{Inv}(D)$ of the invertible ideals of a strongly discrete Pr\"ufer domains is often free: in particular, we show it when the spectrum of $D$ is Noetherian or when $D$ is a ring of integer-valued polynomials on a subset over a Dedekind domain.