Ideal class groups of monoid algebras

TL;DR

This paper investigates the relationship between subintegral closure properties of ring and monoid extensions and the structure of invertible submodules in monoid algebra extensions, providing new characterizations.

Contribution

It establishes an equivalence between subintegral closure of ring and monoid extensions and the isomorphism of groups of invertible submodules in monoid algebra extensions.

Findings

Subintegral closure of rings corresponds to isomorphism of invertible submodule groups.

The result holds for reduced ring extensions and monoid extensions.

For N=N, the reduced assumption on the ring extension is unnecessary.

Abstract

Let $A\subset B$ be an extension of commutative reduced rings and $M\subset N$ an extension of positive commutative cancellative torsion-free monoids. We prove that $A$ is subintegrally closed in $B$ and $M$ is subintegrally closed in $N$ if and only if the group of invertible $A$-submodules of $B$ is isomorphic to the group of invertible $A[M]$-submodules of $B[N]$. In case $M=N$, we prove the same without the assumption that the ring extension is reduced.