On the arithmetic of Krull monoids with infinite cyclic class group

TL;DR

This paper investigates the factorization properties of Krull monoids with infinite cyclic class groups, providing explicit conditions under which key finiteness and structural properties hold.

Contribution

It offers new characterizations of when various finiteness properties are valid in Krull monoids with infinite cyclic class groups based on prime divisor classes.

Findings

Conditions for local tameness established

Finiteness and rationality of elasticity characterized

Structure theorem for sets of lengths derived

Abstract

Let $H$ be a Krull monoid with infinite cyclic class group $G$ and let $G_P \subset G$ denote the set of classes containing prime divisors. We study under which conditions on $G_P$ some of the main finiteness properties of factorization theory--such as local tameness, the finiteness and rationality of the elasticity, the structure theorem for sets of lengths, the finiteness of the catenary degree, and the existence of monotone and of near monotone chains of factorizations--hold in $H$. In many cases, we derive explicit characterizations.