  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.
