Semigroup rings as weakly Krull domains

TL;DR

This paper characterizes when semigroup rings over integral domains are weakly Krull domains based on properties of the domain, the semigroup, and the quotient group, with applications to algebraic structures.

Contribution

It provides a complete characterization of weakly Krull semigroup rings in terms of the properties of the base domain, the semigroup, and the quotient group, including arithmetical applications.

Findings

D[mma] is weakly Krull iff D is a weakly Krull UMT-domain and amma is a weakly Krull UMT-monoid with specific quotient group types.

The characterization depends on the characteristic of the domain, with different conditions for char(D)=0 and char(D)=p>0.

The paper includes arithmetical applications of the main characterization.

Abstract

Let $D$ be an integral domain and $\Gamma$ be a torsion-free commutative cancellative (additive) semigroup with identity element and quotient group $G$. In this paper, we show that if char$(D)=0$ (resp., char$(D)=p>0$), then $D[\Gamma]$ is a weakly Krull domain if and only if $D$ is a weakly Krull UMT-domain, $\Gamma$ is a weakly Krull UMT-monoid, and $G$ is of type $(0,0,0, \dots )$ (resp., type $(0,0,0, \dots )$ except $p$). Moreover, we give arithmetical applications of this result.