  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 $\dim A\leq 1$, 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 $\dim A\geq 2$ 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.
