Semigroup rings as weakly Krull domains

Gyu Whan Chang, Victor Fadinger and Daniel Windisch

Vol. 318 (2022), No. 2, 433–452

DOI: 10.2140/pjm.2022.318.433

Abstract

Let $D$ be an integral domain and $\Gamma$ be a torsion-free commutative cancellative (additive) semigroup with identity element and quotient group $G$. We show that if $\mathrm{char}<!--FUNCTION\; APPLICATION-->(D)=0$ (resp., $\mathrm{char}<!--FUNCTION\; APPLICATION-->(D)=p>0$), 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 $(0,0,0,\dots <!--FUNCTION\; APPLICATION-->)$ (resp., type $(0,0,0,\dots <!--FUNCTION\; APPLICATION-->)$ except $p$). Moreover, we give arithmetical applications of this result.

Keywords

semigroup ring, weakly Krull domain, finite $t$-character, system of sets of lengths

Mathematical Subject Classification

Primary: 13A15, 13F05, 20M12

Milestones

Received: 24 January 2022

Revised: 21 April 2022

Accepted: 21 May 2022

Published: 20 August 2022