Fractional Ideal - Definition and Basic Results

Definition and Basic Results

Let R be an integral domain, and let K be its field of fractions. A fractional ideal of R is an R-submodule I of K such that there exists a non-zero r ∈ R such that rI ⊆ R. The element r can be thought of as clearing out the denominators in I. The principal fractional ideals are those R-submodules of K generated by a single nonzero element of K. A fractional ideal I is contained in R if, and only if, it is an ('integral') ideal of R.

A fractional ideal I is called invertible if there is another fractional ideal J such that IJ = R (where IJ = { a₁b₁ + a₂b₂ + ... + a_nb_n : a_i ∈ I, b_i ∈ J, n ∈ Z_>0 } is called the product of the two fractional ideals). The set of invertible fractional ideals form an abelian group with respect to above product, where the identity is the unit ideal R itself. This group is called the group of fractional ideals of R. The principal fractional ideals form a subgroup. A (nonzero) fractional ideal is invertible if, and only if, it is projective as an R-module.

Every finitely generated R-submodule of K is a fractional ideal and if R is noetherian these are all the fractional ideals of R.