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 ℤ. Let GP⊆G denote the set of classes containing prime divisors. It was shown that ρ(H), the elasticity of H, is finite if and only if GP is bounded above or below. By a result of Lambert, it is easy to show that ρ(H)≤ min{−inf(GP),sup(GP)}. In this paper, we focus on H with large elasticity. When GP is infinite but bounded above or below, we give necessary and sufficient conditions for that ρ(H)> min{−inf(GP),sup(GP)}−32. When GP is a finite set, we give a better upper bound on ρ(H), which is sharp in some sense.