Fractional ideal

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 ${\displaystyle R}$ 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.

Number fields

For the special case of a number field ${\displaystyle K}$ (such as a cyclotomic field) there is an associated ring denoted ${\displaystyle {\mathcal{O}}_K}$ called the ring of integers of ${\displaystyle K}$. For example, ${\displaystyle {\mathcal{O}}_{\mathbb{Q}(\sqrt{d})}=\mathbb{Z}[\sqrt{d}]}$ for ${\displaystyle d}$ square-free and congruent to ${\displaystyle 2,3(\text{mod }4)}$. The key property of these rings ${\displaystyle {\mathcal{O}}_K}$ 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^{pg 2} ${\displaystyle {\mathcal{O}}_K}$ of a number field, the group of fractional ideals forms a group denoted ${\displaystyle {\mathcal{I}}_K}$ and the subgroup of principal fractional ideals is denoted ${\displaystyle {\mathcal{P}}_K}$. The ideal class group is the group of fractional ideals modulo the principal fractional ideals, so

${\displaystyle {\mathcal{C}}_K:={\mathcal{I}}_K/{\mathcal{P}}_K}$

and its class number ${\displaystyle h_K}$ is the order of the group, ${\displaystyle h_K=|{\mathcal{C}}_K|}$. In some ways, the class number is a measure for how "far" the ring of integers ${\displaystyle {\mathcal{O}}_K}$ is from being a unique factorization domain (UFD). This is because ${\displaystyle h_K=1}$ if and only if ${\displaystyle {\mathcal{O}}_K}$ is a UFD.

Exact sequence for ideal class groups

There is an exact sequence

${\displaystyle 0\to {\mathcal{O}}_K^{\ast }\to K^{\ast }\to {\mathcal{I}}_K\to {\mathcal{C}}_K\to 0}$

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 ${\displaystyle I}$ decomposes uniquely up to ordering as

${\displaystyle I=({\mathfrak{p}}_1\dots {\mathfrak{p}}_n)({\mathfrak{q}}_1\dots {\mathfrak{q}}_m{)}^{-1}}$

for prime ideals

${\displaystyle {\mathfrak{p}}_i,{\mathfrak{q}}_j\in \text{Spec}({\mathcal{O}}_K)}$.

in the spectrum of ${\displaystyle {\mathcal{O}}_K}$. For example,

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

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 ${\displaystyle {\mathcal{O}}_K}$ integral.

Examples

• ${\displaystyle \frac{5}{4}\mathbb{Z}}$ is a fractional ideal over ${\displaystyle \mathbb{Z}}$
• For ${\displaystyle K=\mathbb{Q}(i)}$ the ideal ${\displaystyle (5)}$ splits in ${\displaystyle {\mathcal{O}}_{\mathbb{Q}(i)}=\mathbb{Z}[i]}$ as ${\displaystyle (2-i)(2+i)}$
• For ${\displaystyle K={\mathbb{Q}}_{{\zeta }_3}}$ we have the factorization ${\displaystyle (3)=(2{\zeta }_3+1{)}^2}$. This is because if we multiply it out, we get

  ${\displaystyle \begin{array}{cc}(2{\zeta }_3+1{)}^2& =4{\zeta }_3^2+4{\zeta }_3+1\\ & =4({\zeta }_3^2+{\zeta }_3)+1\end{array}}$

Since ${\displaystyle {\zeta }_3}$ satisfies ${\displaystyle {\zeta }_3^2+{\zeta }_3=-1}$, our factorization makes sense.

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

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

to get the ideal

${\displaystyle IJ=\left(\frac{1}{2}\sqrt{-23}+\frac{3}{2}\right).}$

Divisorial ideal

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

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. 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.

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