Editing Fractional ideal (section)

==Definition and basic results==

Let <math>R</math> be an [[integral domain]], and let <math>K = \operatorname{Frac}R</math> be its [[field of fractions]].

A '''fractional ideal''' of <math>R</math> is an <math>R</math>-[[submodule]] <math>I</math> of <math>K</math> such that there exists a non-zero <math>r \in R</math> such that <math>rI\subseteq R</math>. The element <math>r</math> can be thought of as clearing out the denominators in <math>I</math>, hence the name fractional ideal.

The '''principal fractional ideals''' are those <math>R</math>-submodules of <math>K</math> generated by a single nonzero element of <math>K</math>. A fractional ideal <math>I</math> is contained in <math>R</math> [[if and only if]] it is an (integral) ideal of <math>R</math>.

A fractional ideal <math>I</math> is called '''invertible''' if there is another fractional ideal <math>J</math> such that
:<math>IJ = R</math>
where
:<math>IJ = \{ a_1 b_1 + a_2 b_2 + \cdots + a_n b_n : a_i \in I, b_j \in J, n \in \mathbb{Z}_{>0} \}</math>
is the '''product''' of the two fractional ideals.

In this case, the fractional ideal <math>J</math> is uniquely determined and equal to the generalized [[ideal quotient]] 
:<math>(R :_{K} I) = \{ x \in K : xI \subseteq R \}.</math>
The set of invertible fractional ideals form an [[commutative group]] with respect to the above product, where the identity is the [[unit ideal]] <math>(1) = R</math> itself. This group is called the '''group of fractional ideals''' of <math>R</math>. The principal fractional ideals form a [[subgroup]]. A (nonzero) fractional ideal is invertible if and only if it is [[projective module|projective]] as an <math>R</math>-[[module (mathematics)|module]]. Geometrically, this means an invertible fractional ideal can be interpreted as rank 1 [[Vector bundle (algebraic geometry)|vector bundle]] over the [[Spectrum of a ring|affine scheme]] <math>\text{Spec}(R)</math>.

Every [[finitely generated module|finitely generated]] ''R''-submodule of ''K'' is a fractional ideal and if <math>R</math> is [[Noetherian ring|noetherian]] these are all the fractional ideals of <math>R</math>.