On the arithmetic of Mori monoids and domains

Abstract

Let $R$ be a Mori domain with complete integral closure $\hat{R}$, nonzero conductor $\mathfrak{f}=(R:\hat{R})$, and suppose that both $v$-class groups ${\mathcal{C}}_v(R)$ and ${\mathcal{C}}_v(\hat{R})$ are finite. If $R/\mathfrak{f}$ is finite, then the elasticity of $R$ is either rational or infinite. If $R/\mathfrak{f}$ is artinian, then unions of sets of lengths of $R$ are almost arithmetical progressions with the same difference and global bound. We derive our results in the setting of $v$-noetherian monoids.