Editing Fractional ideal

{{Short description|Submodule of fractions in abstract algebra}}
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In [[mathematics]], in particular [[commutative algebra]], the concept of '''fractional ideal''' is introduced in the context of [[integral domain]]s and is particularly fruitful in the study of [[Dedekind domain]]s. In some sense, fractional ideals of an integral domain are like [[ideal (ring theory)|ideal]]s where [[denominator]]s are allowed. In contexts where fractional ideals and ordinary [[ring ideal]]s are both under discussion, the latter are sometimes termed '''''integral ideals''''' for clarity.

==Definition and basic results==

Let <math>R</math> be an [[integral domain]], and let <math>K = \operatorname{Frac}R</math> be its [[field of fractions]].

A '''fractional ideal''' of <math>R</math> is an <math>R</math>-[[submodule]] <math>I</math> of <math>K</math> such that there exists a non-zero <math>r \in R</math> such that <math>rI\subseteq R</math>. The element <math>r</math> can be thought of as clearing out the denominators in <math>I</math>, hence the name fractional ideal.

The '''principal fractional ideals''' are those <math>R</math>-submodules of <math>K</math> generated by a single nonzero element of <math>K</math>. A fractional ideal <math>I</math> is contained in <math>R</math> [[if and only if]] it is an (integral) ideal of <math>R</math>.

A fractional ideal <math>I</math> is called '''invertible''' if there is another fractional ideal <math>J</math> such that
:<math>IJ = R</math>
where
:<math>IJ = \{ a_1 b_1 + a_2 b_2 + \cdots + a_n b_n : a_i \in I, b_j \in J, n \in \mathbb{Z}_{>0} \}</math>
is the '''product''' of the two fractional ideals.

In this case, the fractional ideal <math>J</math> is uniquely determined and equal to the generalized [[ideal quotient]] 
:<math>(R :_{K} I) = \{ x \in K : xI \subseteq R \}.</math>
The set of invertible fractional ideals form an [[commutative group]] with respect to the above product, where the identity is the [[unit ideal]] <math>(1) = R</math> itself. This group is called the '''group of fractional ideals''' of <math>R</math>. The principal fractional ideals form a [[subgroup]]. A (nonzero) fractional ideal is invertible if and only if it is [[projective module|projective]] as an <math>R</math>-[[module (mathematics)|module]]. Geometrically, this means an invertible fractional ideal can be interpreted as rank 1 [[Vector bundle (algebraic geometry)|vector bundle]] over the [[Spectrum of a ring|affine scheme]] <math>\text{Spec}(R)</math>.

Every [[finitely generated module|finitely generated]] ''R''-submodule of ''K'' is a fractional ideal and if <math>R</math> is [[Noetherian ring|noetherian]] these are all the fractional ideals of <math>R</math>.

==Dedekind domains==

In [[Dedekind domain]]s, 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 <math>R</math> is denoted <math>\text{Div}(R)</math>.

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 [[Algebraic number field|number fields]] <math>K</math> (such as <math>\mathbb{Q}(\zeta_n)</math>, where <math>\zeta_n</math> = ''exp(2π i/n)'') there is an associated [[ring (mathematics)|ring]] denoted <math>\mathcal{O}_K</math> called the [[ring of integers]] of <math>K</math>. For example, <math>\mathcal{O}_{\mathbb{Q}(\sqrt{d}\,)} = \mathbb{Z}[\sqrt{d}\,]</math> for <math>d</math> [[squarefree integer|square-free]] and [[modular arithmetic|congruent]] to <math>2,3 \text{ }(\text{mod } 4)</math>. The key property of these rings <math>\mathcal{O}_K</math> 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<ref>{{Cite book|last=Childress|first=Nancy|url=https://www.worldcat.org/oclc/310352143|title=Class field theory|date=2009|publisher=Springer|isbn=978-0-387-72490-4|location=New York|oclc=310352143}}</ref><sup>pg 2</sup> <math>\mathcal{O}_K</math> of a number field, the group of fractional ideals forms a group denoted <math>\mathcal{I}_K</math> and the subgroup of principal fractional ideals is denoted <math>\mathcal{P}_K</math>. The '''[[ideal class group]]''' is the group of fractional ideals modulo the principal fractional ideals, so
: <math>\mathcal{C}_K := \mathcal{I}_K/\mathcal{P}_K</math>
and its class number <math>h_K</math> is the [[order of a group|order]] of the group, <math>h_K = |\mathcal{C}_K|</math>. In some ways, the class number is a measure for how "far" the ring of integers <math>\mathcal{O}_K</math> is from being a [[unique factorization domain]] (UFD). This is because <math>h_K = 1</math> if and only if <math>\mathcal{O}_K</math> is a UFD.

==== Exact sequence for ideal class groups ====
There is an [[exact sequence]]
:<math>0 \to \mathcal{O}_K^* \to K^* \to \mathcal{I}_K \to \mathcal{C}_K \to 0</math>
associated to every number field.

=== Structure theorem for fractional ideals ===
One of the important structure theorems for fractional ideals of a [[number field]] states that every fractional ideal <math>I</math> decomposes uniquely up to ordering as
:<math>I = (\mathfrak{p}_1\ldots\mathfrak{p}_n)(\mathfrak{q}_1\ldots\mathfrak{q}_m)^{-1}</math>
for [[prime ideal]]s
:<math>\mathfrak{p}_i,\mathfrak{q}_j \in \text{Spec}(\mathcal{O}_K)</math>.

in the [[spectrum of a ring|spectrum]] of <math>\mathcal{O}_K</math>. For example,
:<math>\frac{2}{5}\mathcal{O}_{\mathbb{Q}(i)}</math> factors as <math>(1+i)(1-i)((1+2i)(1-2i))^{-1} </math>

Also, because fractional ideals over a number field are all finitely generated we can clear denominators by multiplying by some <math>\alpha</math> to get an ideal <math>J</math>. Hence
: <math>I = \frac{1}{\alpha}J</math>
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 <math>\mathcal{O}_K</math> ''integral''.

==Examples==
* <math>\frac{5}{4}\mathbb{Z}</math> is a fractional ideal over <math>\mathbb{Z}</math>
*For <math>K = \mathbb{Q}(i)</math> the ideal <math>(5)</math> splits in <math>\mathcal{O}_{\mathbb{Q}(i)} = \mathbb{Z}[i]</math> as <math>(2-i)(2+i)</math>
* For <math>K=\mathbb{Q}_{\zeta_3}</math> we have the factorization <math>(3) = (2\zeta_3 + 1)^2</math>. This is because if we multiply it out, we get
*:<math>\begin{align}
(2\zeta_3 + 1)^2 &= 4\zeta_3^2 + 4\zeta_3 + 1 \\
&= 4(\zeta_3^2 + \zeta_3) + 1
\end{align}</math>
:Since <math>\zeta_3</math> satisfies <math>\zeta_3^2 + \zeta_3 =-1</math>, our factorization makes sense.
* For <math>K=\mathbb{Q}(\sqrt{-23})</math> we can multiply the fractional ideals
:: <math>I = \left(2, \frac12\sqrt{-23} - \frac12\right)</math> and <math>J=\left(4,\frac12\sqrt{-23} + \frac32\right)</math>
: to get the ideal
::<math>IJ=\left(\frac12\sqrt{-23}+\frac32\right).</math>

==Divisorial ideal==
Let <math>\tilde I</math> denote the [[intersection (set theory)|intersection]] of all principal fractional ideals containing a nonzero fractional ideal <math>I</math>.

Equivalently,
:<math>\tilde I = (R : (R : I)),</math>
where as above
:<math>(R : I) = \{ x \in K : xI \subseteq R \}. </math> 
If <math>\tilde I = I</math> then ''I'' is called '''divisorial'''.<ref>{{harvnb|Bourbaki|1998|loc=§VII.1}}</ref> 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 ring|local]] [[Krull domain]] (e.g., a [[Noetherian ring|Noetherian]] [[integrally closed domain|integrally closed]] local domain). Then ''R'' is a [[discrete valuation ring]] if and only if the [[maximal ideal]] of ''R'' is divisorial.<ref>{{harvnb|Bourbaki|1998|loc=Ch. VII, § 1, n. 7. Proposition 11.}}</ref>

An integral domain that satisfies the [[ascending chain condition]]s on divisorial ideals is called a [[Mori domain]].{{sfn|Barucci|2000}}

==See also==
*[[Divisorial sheaf]]
*[[Dedekind-Kummer theorem]]

==Notes==
{{reflist}}

==References==
*{{Citation | last1=Barucci | first1=Valentina | editor1-last=Glaz | editor1-first=Sarah|editor1-link=Sarah Glaz | editor2-last=Chapman | editor2-first=Scott T. | title=Non-Noetherian commutative ring theory | chapter-url=https://books.google.com/books?id=0tuZkZE07TEC | publisher=Kluwer Acad. Publ. | location=Dordrecht | series=Mathematics and its Applications | isbn=978-0-7923-6492-4  |mr=1858157 | year=2000 | volume=520 | chapter=Mori domains | pages=57–73}}
* {{Citation | last=Stein | first=William | title=A Computational Introduction to Algebraic Number Theory | url=http://wstein.org/books/ant/ant.pdf }}
*Chapter 9 of {{Citation | last1=Atiyah | first1=Michael Francis | author1-link=Michael Atiyah | last2=Macdonald | first2=I.G. | author2-link=Ian G. Macdonald | title=Introduction to Commutative Algebra | publisher=Westview Press | isbn=978-0-201-40751-8 | year=1994}}
*Chapter VII.1 of {{Citation | last1=Bourbaki | first1=Nicolas | author1-link=Nicolas Bourbaki | title=Commutative algebra | publisher=[[Springer Verlag]] | edition=2nd | year=1998 | isbn=3-540-64239-0 }}
*Chapter 11 of {{Citation | last1=Matsumura | first1=Hideyuki | title=Commutative ring theory | publisher=[[Cambridge University Press]] | edition=2nd | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-36764-6 | mr=1011461  | year=1989 | volume=8}}

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[[Category:Ideals (ring theory)]]
[[Category:Algebraic number theory]]