
Fractional ideal

In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity.
Definition and basic results.
Let formula_1 be an integral domain, and let formula_2 be its field of fractions.
A fractional ideal of formula_1 is an formula_1-submodule formula_5 of formula_6 such that there exists a non-zero formula_7 such that formula_8. The element formula_9 can be thought of as clearing out the denominators in formula_5, hence the name fractional ideal.
The principal fractional ideals are those formula_1-submodules of formula_6 generated by a single nonzero element of formula_6. A fractional ideal formula_5 is contained in formula_1 if, and only if, it is an ('integral') ideal of formula_1.
A fractional ideal formula_5 is called invertible if there is another fractional ideal formula_18 such thatformula_19whereformula_20is called the product of the two fractional ideals).
In this case, the fractional ideal formula_18 is uniquely determined and equal to the generalized ideal quotient 
The set of invertible fractional ideals form an abelian group with respect to the above product, where the identity is the unit ideal formula_23 itself. This group is called the group of fractional ideals of formula_1. The principal fractional ideals form a subgroup. A (nonzero) fractional ideal is invertible if, and only if, it is projective as an formula_1-module. Geometrically, this means an invertible fractional ideal can be interpreted as rank 1 vector bundles over the affine scheme formula_26. 
Every finitely generated "R"-submodule of "K" is a fractional ideal and if formula_1 is noetherian these are all the fractional ideals of formula_1.
Dedekind domains.
In Dedekind domains, the situation is much simpler. In particular, every non-zero fractional ideal is invertible. In fact, this property characterizes Dedekind domains:
The set of fractional ideals over a Dedekind domain formula_1 is denoted formula_30.
Its quotient group of fractional ideals by the subgroup of principal fractional ideals is an important invariant of a Dedekind domain called the ideal class group.
Number fields.
For the special case of number fields formula_6 (such as formula_32) there is an associated ring denoted formula_33 called the ring of integers of formula_6. For example, formula_35 for formula_36 square free and equal to formula_37. The key property of these rings formula_33 is they are Dedekind domains. Hence the theory of fractional ideals can be described for the rings of integers of number fields. In fact, class field theory is the study of such groups of class rings.
Associated structures.
For the ring of integerspg 2 formula_33 of a number field, the group of fractional ideals forms a group denoted formula_40 and the subgroup of principal fractional ideals is denoted formula_41. The ideal class group is the group of fractional ideals modulo the principal fractional ideals, soformula_42and its class number formula_43 is the order of the group formula_44. In some ways, the class number is a measure for how "far" the ring of integers formula_33 is from being a unique factorization domain. This is because formula_46 if and only if formula_33 is a UFD.
Exact sequence for ideal class groups.
There is an exact sequence
associated to every number field.
Structure theorem for fractional ideals.
One of the important structure theorems for fractional ideals of a number field states that every fractional ideal formula_5 decomposes uniquely up to ordering as
for prime ideals
in the spectrum of formula_33. For example,
Also, because fractional ideals over a number field are all finitely generated we can clear denominators by multiplying by some formula_55 to get an ideal formula_18. Hence
formula_57
Another useful structure theorem is that integral fractional ideals are generated by up to 2 elements. We call a fractional ideal which is a subset of formula_33 integral.
Divisorial ideal.
Let formula_77 denote the intersection of all principal fractional ideals containing a nonzero fractional ideal formula_5.
Equivalently,
where as above
If formula_81 then "I" is called divisorial.
In other words, a divisorial ideal is a nonzero intersection of some nonempty set of fractional principal ideals.
If "I" is divisorial and "J" is a nonzero fractional ideal, then ("I" : "J") is divisorial.
Let "R" be a local Krull domain (e.g., a Noetherian integrally closed local domain).
Then "R" is a discrete valuation ring if and only if the maximal ideal of "R" is divisorial.
An integral domain that satisfies the ascending chain conditions on divisorial ideals is called a Mori domain.