Algebraic structure → Ring theory Ring theory |
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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.

Let be an integral domain, and let be its field of fractions.

A **fractional ideal** of is an -submodule of such that there exists a non-zero such that . The element can be thought of as clearing out the denominators in , hence the name fractional ideal.

The **principal fractional ideals** are those -submodules of generated by a single nonzero element of . A fractional ideal is contained in if and only if it is an (integral) ideal of .

A fractional ideal is called **invertible** if there is another fractional ideal such that

where

is the **product** of the two fractional ideals.

In this case, the fractional ideal 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 itself. This group is called the **group of fractional ideals** of . The principal fractional ideals form a subgroup. A (nonzero) fractional ideal is invertible if and only if it is projective as an -module. Geometrically, this means an invertible fractional ideal can be interpreted as rank 1 vector bundle over the affine scheme .

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

In Dedekind domains, the situation is much simpler. In particular, every non-zero fractional ideal is invertible. In fact, this property characterizes Dedekind domains:

- An integral domain is a Dedekind domain if and only if every non-zero fractional ideal is invertible.

The set of fractional ideals over a Dedekind domain is denoted .

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.

For the special case of number fields (such as ) there is an associated ring denoted called the ring of integers of . For example, for square-free and congruent to . The key property of these rings 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.

For the ring of integers^{[1]}^{pg 2} of a number field, the group of fractional ideals forms a group denoted and the subgroup of principal fractional ideals is denoted . The **ideal class group** is the group of fractional ideals modulo the principal fractional ideals, so

and its class number is the order of the group, . In some ways, the class number is a measure for how "far" the ring of integers is from being a unique factorization domain (UFD). This is because if and only if is a UFD.

There is an exact sequence

associated to every number field.

One of the important structure theorems for fractional ideals of a number field states that every fractional ideal decomposes uniquely up to ordering as

for prime ideals

- .

in the spectrum of . For example,

- factors as

Also, because fractional ideals over a number field are all finitely generated we can clear denominators by multiplying by some to get an ideal . Hence

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 *integral*.

- is a fractional ideal over
- For the ideal splits in as
- In we have the factorization . This is because if we multiply it out, we get

- Since satisfies , our factorization makes sense.

- In we can multiply the fractional ideals

- and

- to get the ideal

Let denote the intersection of all principal fractional ideals containing a nonzero fractional ideal .

Equivalently,

where as above

If then *I* is called **divisorial**.^{[2]} 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.^{[3]}

An integral domain that satisfies the ascending chain conditions on divisorial ideals is called a Mori domain.^{[4]}