On the arithmetic of monoids of ideals

TL;DR

This paper explores the algebraic and arithmetic properties of monoids of invertible ideals in Krull and Mori domains, revealing their similarities to Krull monoids with infinite class groups despite not being transfer Krull.

Contribution

It provides new insights into the structure and arithmetic of monoids of ideals in specific algebraic domains, including polynomial rings over noetherian domains.

Findings

Monoids of invertible ideals in Krull and Mori domains exhibit specific arithmetical phenomena.

Polynomial rings with multiple indeterminates share properties with Krull monoids.

These monoids are not transfer Krull but display similar arithmetical behavior.

Abstract

We study the algebraic and arithmetic structure of monoids of invertible ideals (more precisely, of $r$-invertible $r$-ideals for certain ideal systems $r$) of Krull and weakly Krull Mori domains. We also investigate monoids of all nonzero ideals of polynomial rings with at least two indeterminates over noetherian domains. Among others, we show that they are not transfer Krull but they share several arithmetical phenomena with Krull monoids having infinite class group and prime divisors in all classes.