Elasticities of Krull monoids with infinite cyclic class group

Abstract

Let $H$ be a Krull monoid with infinite cyclic class group $G$ that we identify with $\mathbb{Z}$. Let $G_P\subseteq G$ denote the set of classes containing prime divisors. It was shown that $\rho \left(H\right)$, the elasticity of $H$, is finite if and only if $G_P$ is bounded above or below. By a result of Lambert, it is easy to show that $\rho \left(H\right)\le min\left\{-inf\left(G_P\right),sup\left(G_P\right)\right\}$. In this paper, we focus on $H$ with large elasticity. When $G_P$ is infinite but bounded above or below, we give necessary and sufficient conditions for that . When $G_P$ is a finite set, we give a better upper bound on $\rho \left(H\right)$, which is sharp in some sense.