In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in particular commutative algebra

Commutative algebra

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...

, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domain

Dedekind domain

In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...

s. 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 K be its field of fractions

Field of fractions

In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...

. 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

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

 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

Projective module

In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...

 as an R-module.

Every finitely generated R-submodule of K is a fractional ideal and if R is noetherian

Noetherian ring

In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...

 these are all the fractional ideals of R.

Dedekind domains

In Dedekind domains, the situation is much nicer. 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 quotient group

Quotient group

In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

 of fractional ideals by the subgroup of principal fractional ideals is an important invariant of a Dedekind domain called the ideal class group

Ideal class group

In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...

. Part of the reason for introducing fractional ideals is to realize the ideal class group as an actual quotient group, rather than with the ad hoc multiplication of equivalence classes of ideals.

Other notions

Let denote the intersection of all principal fractional ideals containing I. Equivalently,
where.
If then I is called divisorial.