Sets of Arithmetical Invariants in Transfer Krull Monoids

TL;DR

This paper explores the arithmetical invariants of transfer Krull monoids, demonstrating their full elasticity and characterizing the structure of catenary and tame degrees under various conditions.

Contribution

It establishes that transfer Krull monoids are fully elastic and characterizes the sets of catenary and tame degrees, including their possible configurations.

Findings

Transfer Krull monoids are fully elastic.

Sets of catenary and tame degrees can be intervals or arbitrary finite sets.

Results apply to both commutative and certain non-commutative domains.

Abstract

Transfer Krull monoids are a recent concept including all commutative Krull domains and also, for example, wide classes of non-commutative Dedekind domains. We show that transfer Krull monoids are fully elastic (i.e., every rational number between $1$ and the elasticity of the monoid can be realized as the elasticity of an element). In commutative Krull monoids which have sufficiently many prime divisors in all classes of their class group, the set of catenary degrees and the set of tame degrees are intervals. Without the assumption on the distribution of prime divisors, arbitrary finite sets can be realized as sets of catenary degrees and as sets of tame degrees.