Fractional ideal

Ring homomorphisms Algebraic structures Related structures Algebraic number theory Noncommutative algebraic geometry Free algebra Clifford algebra 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.

generated by a single nonzero element of

is called invertible if there is another 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

The principal fractional ideals form a subgroup.

A (nonzero) fractional ideal is invertible if and only if it is projective as an

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

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

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

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.

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

In some ways, the class number is a measure for how "far" the ring of integers

is from being a unique factorization domain (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, Also, because fractional ideals over a number field are all finitely generated we can clear denominators by multiplying by some

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

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

[2] In other words, a divisorial ideal is a nonzero intersection of some nonempty set of fractional principal ideals.

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.