Fractional ideal

Summary

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 resultsEdit

Let   be an integral domain, and let   be its field of fractions.

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 called 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 bundles over the affine scheme  .

Every finitely generated R-submodule of K is a fractional ideal and if   is noetherian these are all the fractional ideals of  .

Dedekind domainsEdit

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 fieldsEdit

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 equal 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 structuresEdit

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. This is because   if and only if   is a UFD.

Exact sequence for ideal class groupsEdit

There is an exact sequence

 

associated to every number field.

Structure theorem for fractional idealsEdit

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,

  factors 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.

ExamplesEdit

  •   is a fractional ideal over  
  • For   the ideal   splits in   as  
  • In   we have the factorization  .
This is because if we multiply it out, we get
 
Since   satisfies  , our factorization makes sense.
  • In   we can multiply the fractional ideals
  •   and
  •  
to get the ideal
 

Divisorial idealEdit

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.

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.[4]

See alsoEdit

NotesEdit

  1. ^ Childress, Nancy (2009). Class field theory. New York: Springer. ISBN 978-0-387-72490-4. OCLC 310352143.
  2. ^ Bourbaki 1998, §VII.1
  3. ^ Bourbaki & Ch. VII, § 1, n. 7. Proposition 11.
  4. ^ http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdffirstpage_1&handle=euclid.rmjm/1187453107

ReferencesEdit