Ideal class groups of monoid algebras

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.

arXiv commentTitle is changed. Main theorem is improved significantly, viz, relative version of monoid is considered. Final version is going to appear in Journal of Commutative Algebra