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 results

edit

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

Dedekind domains

edit

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

edit

For the special case of number fields   (such as  , where   = exp(2π i/n)) 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

edit

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.

Exact sequence for ideal class groups

edit

There is an exact sequence

 

associated to every number field.

Structure theorem for fractional ideals

edit

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.

Examples

edit
  •   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

edit

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 also

edit

Notes

edit
  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 1998, Ch. VII, § 1, n. 7. Proposition 11.
  4. ^ Barucci 2000.

References

edit
  • 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, ISBN 978-0-7923-6492-4, MR 1858157
  • Stein, William, A Computational Introduction to Algebraic Number Theory (PDF)
  • Chapter 9 of Atiyah, Michael Francis; Macdonald, I.G. (1994), Introduction to Commutative Algebra, Westview Press, ISBN 978-0-201-40751-8
  • Chapter VII.1 of Bourbaki, Nicolas (1998), Commutative algebra (2nd ed.), Springer Verlag, ISBN 3-540-64239-0
  • Chapter 11 of Matsumura, Hideyuki (1989), Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8 (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6, MR 1011461