= Invertible fractional ideal
{title2=$II^{-1}=A$}
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 $1=\sum u_iv_i$ with $u_i\in I$, $v_i\in J$. On the distinguished open where $u_iv_i$ is invertible, $I$ is generated by $u_i$, since $x/u_i=(xv_i)/(u_iv_i)$. The maps $x\mapsto v_ix$ 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 $II^{-1}=A$.
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