Subintegrality and ideal class groups of monoid algebras

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 $\frac{B[N]}{IB[N]}$.