A characterization of weakly Krull monoid algebras

Victor Fadinger,Daniel Windisch

doi:10.1016/j.jalgebra.2021.10.013

Abstract

Let D be a domain and let S be a torsion-free monoid such that D has characteristic 0 or the quotient group of S satisfies the ascending chain condition on cyclic subgroups. We give a characterization of when the monoid algebra D[S] is weakly Krull. As corollaries, we reobtain the results on when D[S] is Krull resp. weakly factorial, due to Chouinard resp. Chang. Furthermore, we deduce a characterization of generalized Krull monoid algebras analogous to our main result and we characterize the weakly Krull domains among the affine monoid algebras.

Highlights

When considering monoid algebras D[S], an immediate question is whether certain properties of the domain D and the monoid S carry over to the monoid algebra and
A lot of such properties are studied in the textbook by Gilmer [15], among them the property of being a Krull domain. This property is completely characterized in terms of the domain and the monoid, and this characterization was originally proved by Chouinard [9]. There he was the first to give the definition of a Krull monoid, which in turn was preparing the ground for a whole new research area based on the fact that a domain is Krull if and only if its multiplicative monoid is a Krull monoid
A divisor-theoretic characterization of weakly Krull monoids was first given by Halter-Koch in [19], where one can find a proof of the statement that a domain is weakly Krull if and only if its multiplicative monoid is

Summary

Introduction

When considering monoid algebras D[S], an immediate question is whether certain properties of the domain D and the monoid S carry over to the monoid algebra and . An immediate consequence of the results by Chouinard, and El Baghdadi and Kim is the fact that the polynomial ring over a (generalized) Krull domain is (generalized) Krull and one would hope so as well for the case of weakly Krull domains That this is not the case was proven by D.D. Anderson, Houston and Zafrullah in [2, Proposition 4.11]. Answering Chang’s question and trying to close the last gap for a complete picture of the generalizations of Krull domains, in Section 3, we give a characterization of when a monoid algebra is weakly Krull under the assumption that the quotient group of the monoid satisfies the ACC on cyclic subgroups. We reobtain the results by Chouinard, El Baghdadi and Kim, and Chang

Preliminaries

Main Result

Applications