 Invertible fractional ideal

Over an integral domain $A$ with fraction field $K$, an invertible fractional ideal is a finitely generated nonzero $A$-submodule $I\subseteq K$ for which another fractional ideal $J$ satisfies $IJ=A$. Here $I^{-1}=\{x\in K:xI\subseteq A\}$ equals $J$. Choose $1=\sum u_i{v}_i$ with $u_i\in I$, $v_i\in J$. On the distinguished open where $u_i{v}_i$ is invertible, $I$ is generated by $u_i$, since $x/u_i=(x{v}_i)/(u_i{v}_i)$. The maps $x\mapsto v_{i}x$ form a finite dual basis and prove that $I$ is a rank-one projective module. Conversely, a finitely generated rank-one projective module embeds in $K$ after choosing a basis over $K$; its dual identifies with $I^{-1}$, and local evaluation shows $I{I}^{-1}=A$.