Subintegrality and ideal class groups of monoid algebras

TL;DR

This paper investigates the conditions under which subintegrality notions coincide in monoid algebra extensions and explores the ideal class groups related to these algebraic structures, providing new insights into their properties.

Contribution

It establishes the equivalence of subintegral and weakly subintegral notions in monoid algebra extensions under certain conditions and characterizes subintegrally closed subrings via invertible module groups.

Findings

Subintegral and weakly subintegral notions coincide when $bZ ot i A$.

Characterization of subintegrally closed subrings via invertible module groups.

Conditions for subintegral closure in monoid algebra extensions.

Abstract

$(1)$ Let $M\subset N$ be a commutative cancellative torsion-free and subintegral extension of monoids. Then we prove that in the case of ring extension $A[M]\subset A[N]$, the two notions, subintegral and weakly subintegral coincide provided $\mathbb{Z}\subset A$. $(2)$ Let $A \subset B$ be an extension of commutative rings and $M\subset N$ an extension of commutative cancellative torsion-free positive monoids. Let $I$ be a radical ideal in $N$. Then $\frac{A[M]}{(I\cap M)A[M]}$ is subintegrally closed in $\frac{B[N]}{IB[N]}$ if and only if the group of invertible $A$-submodules of $B$ is isomorphic to the group of invertible $\frac{A[M]}{(I\cap M)A[M]}$-submodules of $\frac{B[N]}{IB[N]}$.