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 idealss 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

be an integral domain be its field of fractions.

A fractional ideal-submodule, hence the name fractional ideal.

The principal fractional ideals if and only if.

is called invertible such that

where

is the product of the two fractional ideals.

is uniquely determined and equal to the generalized ideal quotient

The set of invertible fractional ideals forms a commutative group with respect to the above product, where the identity is the unit ideal itself. This group is called the group of fractional ideals. The principal fractional ideals form a subgroup. A (nonzero) fractional ideal is invertible if and only if it is projective-module. Geometrically, this means an invertible fractional ideal can be interpreted as a rank 1 vector bundle over the affine scheme.

Every finitely generated R-submodule of K is noetherian.

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:

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

.

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 a number field (such as a cyclotomic field) there is an associated ring called the ring of integers square-free and congruent 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. The ideal class group is the group of fractional ideals modulo the principal fractional ideals, so

is the order is from being a unique factorization domain 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 decomposes uniquely up to ordering as

for prime ideals

1. .

in the spectrum. For example,

integral.

Examples

• . This is because if we multiply it out, we get

• , our factorization makes sense.

• we can multiply the fractional ideals

• to get the ideal

Divisorial ideal

denote the intersection.

Equivalently,

where as above

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.

See also

• Divisorial sheaf

• Dedekind–Kummer theorem

Notes

References

• Chapter 9 of

• Chapter VII.1 of

• Chapter 11 of