On the arithmetic of Mori monoids and domains

Abstract

Let $R$ be a Mori domain with complete integral closure $\widehat R$, nonzero conductor $\mathfrak f = (R \ :\ \widehat R)$, and suppose that both $v$-class groups $\mathcal C_v (R)$ and $\mathcal C_v (\widehat R)$ 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.