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.
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 generatedR-submodule of K is a fractional ideal and if is
noetherian these are all the fractional ideals of .
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:
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.
Number fields
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.
Associated structures
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.
One of the important structure theorems for fractional ideals of a
number field states that every fractional ideal decomposes uniquely up to ordering 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.
Examples
is a fractional ideal over
For the ideal splits in as
For we have the factorization . This is because if we multiply it out, we get
Since satisfies , our factorization makes sense.
For we can multiply the fractional ideals
and
to get the ideal
Divisorial 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.
Barucci, Valentina (2000),
"Mori domains", in
Glaz, Sarah; Chapman, Scott T. (eds.), Non-Noetherian commutative ring theory, Mathematics and its Applications, vol. 520, Dordrecht: Kluwer Acad. Publ., pp. 57–73,
ISBN978-0-7923-6492-4,
MR1858157