Fractional ideal ($I\subseteq Q(A)$)

For an integral domain $A$ with fraction field $K$, a fractional ideal is a nonzero $A$-submodule $I\subseteq K$ for which $dI\subseteq A$ for some $0\ne d\in A$. A finitely generated submodule of $K$ has a common denominator and is therefore a fractional ideal. The product is generated by pairwise products of elements. An invertible fractional ideal has a multiplicative inverse and is finitely generated; arbitrary fractional ideals need not be finitely generated over a non-Noetherian domain. The familiar fractional ideals of a number field are the special case $A={\mathcal{O}}_K$. This general definition also applies to the ideals representing Cartier divisors on affine integral schemes.

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 $I{I}^{-1}=A$.

A principal fractional ideal of an integral domain $A$ with fraction field $K$ is $aA$ for a nonzero $a\in K$. Its generators differ by a unit in a ring in $A$. For a number field, taking $A={\mathcal{O}}_K$ and quotienting nonzero fractional ideals by principal fractional ideals gives the ideal class group; requiring congruences and signs of the generators instead gives a ray class group.

An integral ideal is a fractional ideal of an integral domain $A$ that is contained in $A$. For a number field, take $A$ to be its ring of integers ${\mathcal{O}}_K$.