Fractional ideal

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In

are both under discussion, the latter are sometimes termed integral ideals for clarity.

Let ${\displaystyle R}$ be an integral domain, and let ${\displaystyle K=\operatorname {Frac} R}$ be its field of fractions.

A fractional ideal of ${\displaystyle R}$ is an ${\displaystyle R}$-

${\displaystyle I}$ of ${\displaystyle K}$ such that there exists a non-zero ${\displaystyle r\in R}$ such that ${\displaystyle rI\subseteq R}$. The element ${\displaystyle r}$ can be thought of as clearing out the denominators in ${\displaystyle I}$, hence the name fractional ideal.

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

A fractional ideal ${\displaystyle I}$ is called invertible if there is another fractional ideal ${\displaystyle J}$ such that

${\displaystyle IJ=R}$

where

${\displaystyle IJ=\{a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}:a_{i}\in I,b_{j}\in J,n\in \mathbb {Z} _{>0}\}}$

is the product of the two fractional ideals.

In this case, the fractional ideal ${\displaystyle J}$ is uniquely determined and equal to the generalized ideal quotient

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

The set of invertible fractional ideals form an

${\displaystyle (1)=R}$ itself. This group is called the group of fractional ideals of ${\displaystyle R}$. The principal fractional ideals form a subgroup. A (nonzero) fractional ideal is invertible if and only if it is projective as an ${\displaystyle R}$- ${\displaystyle {\text{Spec}}(R)}$.

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

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.

For the special case of number fields ${\displaystyle K}$ (such as ${\displaystyle \mathbb {Q} (\zeta _{n})}$, where ${\displaystyle \zeta _{n}}$ = exp(2π i/n)) 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 ${\displaystyle d}$

to ${\displaystyle 2,3{\text{ }}({\text{mod }}4)}$. 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.

For the ring of integers^{[1]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

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.

There is an exact sequence

${\displaystyle 0\to {\mathcal {O}}_{K}^{*}\to K^{*}\to {\mathcal {I}}_{K}\to {\mathcal {C}}_{K}\to 0}$

associated to every number field.

One of the important structure theorems for fractional ideals of a

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

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

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

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

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

• ${\displaystyle {\frac {5}{4}}\mathbb {Z} }$ 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

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

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

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

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

to get the ideal

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

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.^[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

An integral domain that satisfies the ascending chain conditions on divisorial ideals is called a Mori domain.^[4]

• Divisorial sheaf
• Dedekind-Kummer theorem

References

• Barucci, Valentina (2000), "Mori domains", in
  MR 1858157
• Stein, William, A Computational Introduction to Algebraic Number Theory (PDF)
• Chapter 9 of
  ISBN 978-0-201-40751-8
• Chapter VII.1 of
  ISBN 3-540-64239-0
• Chapter 11 of Matsumura, Hideyuki (1989), Commutative ring theory, Cambridge Studies in Advanced Mathematics, vol. 8 (2nd ed.),
  MR 1011461