Fractional ideal

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

A fractional ideal of R is an R-submodule I of K such that there exists a non-zero r ∈ R such that rI ⊆ R. The element r can be thought of as clearing out the denominators in I.

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

A fractional ideal I is called invertible if there is another fractional ideal J such that

IJ = R

(where IJ = { a₁b₁ + a₂b₂ + ... + a_nb_n : a_i ∈ I, b_i ∈ J, n ∈ Z_>0 } is called the product of the two fractional ideals).

In this case, the fractional ideal J is uniquely determined and equal to the generalized ideal quotient

${\displaystyle (R:I)=\{x\in K:xI\subseteq R\}.}$

The set of invertible fractional ideals form an abelian group with respect to the above product, where the identity is the unit ideal R itself. This group is called the group of fractional ideals of R. The principal fractional ideals form a subgroup. A (nonzero) fractional ideal is invertible if, and only if, it is projective as an R-module.

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

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 ${\displaystyle R}$ is denoted ${\displaystyle {\text{Div}}(R)}$.

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.

Recall that the ring of integers ${\displaystyle {\mathcal {O}}_{K}}$ of a number field ${\displaystyle K}$ is a Dedekind domain.

We call a fractional ideal which is a subset of ${\displaystyle {\mathcal {O}}_{K}}$ integral.

One of the important structure theorems for fractional ideals of a number field states that every fractional ideal ${\displaystyle I}$ decomposes uniquely up to ordering as

${\displaystyle I=({\mathfrak {p}}_{1}\ldots {\mathfrak {p}}_{n})({\mathfrak {q}}_{1}\ldots {\mathfrak {q}}_{m})^{-1}}$

for prime ideals

${\displaystyle {\mathfrak {p}}_{i},{\mathfrak {q}}_{j}\in {\text{Spec}}({\mathcal {O}}_{K})}$.

For example,

${\displaystyle {\frac {2}{5}}{\mathcal {O}}_{\mathbb {Q} (i)}}$ factors as ${\displaystyle (1+i)(1-i)((1+2i)(1-2i))^{-1}}$

Also, because fractional ideals over a number field are all finitely generated we can clear denominators by multiplying by some ${\displaystyle \alpha }$ to get an ideal ${\displaystyle J}$. Hence

${\displaystyle I={\frac {1}{\alpha }}J}$

Another useful structure theorem is that integral fractional ideals are generated by up to 2 elements.

There is an exact sequence

${\displaystyle 0\to {\mathcal {O}}_{K}^{*}\to K^{*}\to {\text{Div}}({\mathcal {O}}_{K})\to C_{K}\to 0}$

associated to every number field,

where

${\displaystyle C_{K}}$ is the ideal class group of ${\displaystyle {\mathcal {O}}_{K}}$.

• ${\displaystyle {\frac {5}{4}}\mathbb {Z} }$ is a fractional ideal over ${\displaystyle \mathbb {Z} }$

• In ${\displaystyle \mathbb {Q} _{\zeta _{3}}}$ we have the factorization ${\displaystyle (3)=(2\zeta _{3}+1)^{2}}$.

This is because if we multiply it out, we get

${\displaystyle {\begin{aligned}(2\zeta _{3}+1)^{2}&=4\zeta _{3}^{2}+4\zeta _{3}+1\\&=4(\zeta _{3}^{2}+\zeta _{3})+1\end{aligned}}}$

Since ${\displaystyle \zeta _{3}}$ satisfies ${\displaystyle \zeta _{3}^{2}+\zeta _{3}=-1}$, our factorization makes sense.

• In ${\displaystyle \mathbb {Q} ({\sqrt {-23}})}$ we can multiply the fractional ideals

• ${\displaystyle I=(2,(1/2){\sqrt {-23}}-(1/2))}$ and
• ${\displaystyle J=(4,(1/2){\sqrt {-23}}+(3/2))}$

to get the ideal

${\displaystyle IJ=(-(1/2){\sqrt {-23}}-(3/2))}$

Let ${\displaystyle {\tilde {I}}}$ denote the intersection of all principal fractional ideals containing a nonzero fractional ideal I.

Equivalently,

${\displaystyle {\tilde {I}}=(R:(R:I)),}$

where as above

${\displaystyle (R:I)=\{x\in K:xI\subseteq R\}.}$

If ${\displaystyle {\tilde {I}}=I}$ then I is called divisorial.[1]

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.[2]

An integral domain that satisfies the ascending chain conditions on divisorial ideals is called a Mori domain.[3]

• Divisorial sheaf

• 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, 8 (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6, MR 1011461