Subintegrality, Invertible Modules and Laurent Polynomial Extensions

Let $A\subseteq B$ be a commutative ring extension. Let $\mathcal{I}(A,B)$ be the multiplicative group of invertible $A$-submodules of $B$. In this article, we extend a result of Sadhu and Singh by finding a necessary and sufficient condition on an integral birational extension $A\subseteq B$ of integral domains with $\dim A\le 1$, so that the natural map $\mathcal I(A,B) \rightarrow \mathcal I (A [X, X^{-1}],B [X, X^{-1}])$ is an isomorphism. In the same situation, we show that if $\dim A\ge 2$ then the condition is necessary but not sufficient. We also discuss some properties of the cokernel of the natural map $\mathcal I(A,B) \rightarrow \mathcal I (A [X, X^{-1}],B [X, X^{-1}])$ in the general case.