The monotone catenary degree of monoids of ideals

TL;DR

This paper investigates the factorization properties of monoids of ideals in integral domains, focusing on the monotone catenary degree and the structure of sets of lengths under certain finiteness conditions.

Contribution

It introduces new finiteness results for the monotone catenary degree and the structure of sets of lengths in monoids of ideals in weakly Krull Mori domains.

Findings

Finiteness results for the monotone catenary degree.

Structural insights into sets of lengths and their unions.

Applicability to monoids of divisorial and v-invertible ideals.

Abstract

Factoring ideals in integral domains is a central topic in multiplicative ideal theory. In the present paper we study monoids of ideals and consider factorizations of ideals into multiplicatively irreducible ideals. The focus is on the monoid of nonzero divisorial ideals and on the monoid of $v$-invertible divisorial ideals in weakly Krull Mori domains. Under suitable algebraic finiteness conditions we establish arithmetical finiteness results, in particular for the monotone catenary degree and for the structure of sets of lengths and of their unions.