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 R be an integral domain, and let 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 \in R such that. The element r can be thought of as clearing out the denominators in I, hence the name fractional ideal. 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 where is the product of the two fractional ideals. In this case, the fractional ideal J 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 (1) = 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. 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 R is noetherian these are all the fractional ideals of R.

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 R 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.

Number fields

For the special case of number fields K (such as, where \zeta_n = exp(2π i/n)) there is an associated ring denoted called the ring of integers of K. For example, for d 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.

Associated structures

For the ring of integers 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 h_K 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 h_K = 1 if and only if 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 I decomposes uniquely up to ordering as for prime ideals in the spectrum of. For example, Also, because fractional ideals over a number field are all finitely generated we can clear denominators by multiplying by some \alpha to get an ideal J. 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.

Examples

Divisorial ideal

Let \tilde I denote the intersection of all principal fractional ideals containing a nonzero fractional ideal I. Equivalently, where as above If 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.