Fractional ideal

 ( Ideals ) 

In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of and is particularly fruitful in the study of . In some sense, fractional ideals of an integral domain are like ideals where are allowed. In contexts where fractional ideals and ordinary are both under discussion, the latter are sometimes termed integral ideals for clarity.

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Definition and basic results Let $R$ be an integral domain, and let $K\; =\; \backslash operatorname\{Frac\}R$ be its field of fractions.

A fractional ideal of $R$ is an $R$-submodule $I$ of $K$ such that there exists a non-zero $r\; \backslash in\; R$ such that $rI\backslash subseteq\; R$. Equivalently, $I\; \backslash subseteq\; K$ is a fractional ideal of $R$ if $I\; =\; r^\{-1\}J$, where $r$ is a non-zero element of $R$ and $J$ is an ideal of $R$. The element $r$ can be thought of as clearing out the denominators in $I$, hence the name fractional ideal.

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

$IJ\; =\; \backslash \{\; a\_1\; b\_1\; +\; a\_2\; b\_2\; +\; \backslash cdots\; +\; a\_n\; b\_n\; :\; a\_i\; \backslash in\; I,\; b\_j\; \backslash in\; J,\; n\; \backslash in\; \backslash mathbb\{Z\}\_\{>0\}\; \backslash \}$

is the product of the two fractional ideals.

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

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

The set of invertible fractional ideals forms a commutative group with respect to the above product, where the identity is the unit ideal $(1)\; =\; R$ 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. Geometrically, this means an invertible fractional ideal can be interpreted as a rank 1 vector bundle over the affine scheme $\backslash text\{Spec\}(R)$.

Every finitely generated R-submodule of K is a fractional ideal and if $R$ is Noetherian ring, then these are all the fractional ideals of $R$.

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Dedekind domains In , 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 $R$ is denoted $\backslash text\{Div\}(R)$.

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.

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Number fields For the special case of a number field $K$ (such as a cyclotomic field) there is an associated ring denoted $\backslash mathcal\{O\}\_K$ called the ring of integers of $K$. For example, $\backslash mathcal\{O\}\_\{\backslash mathbb\{Q\}(\backslash sqrt\{d\}\backslash ,)\}\; =\; \backslash mathbb\{Z\}\backslash sqrt\{d\}\backslash ,$ for $d$ square-free and congruent to $2,3\; \backslash text\{\; \}(\backslash text\{mod\; \}\; 4)$. The key property of these rings $\backslash mathcal\{O\}\_K$ 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.

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Associated structures For the ring of integers

Nancy Childress (2026). 9780387724904, Springer. ISBN 9780387724904

^{pg 2} $\backslash mathcal\{O\}\_K$ of a number field, the group of fractional ideals forms a group denoted $\backslash mathcal\{I\}\_K$ and the subgroup of principal fractional ideals is denoted $\backslash mathcal\{P\}\_K$. The ideal class group is the group of fractional ideals modulo the principal fractional ideals, so

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

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

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Exact sequence for ideal class groups There is an exact sequence

$0\; \backslash to\; \backslash mathcal\{O\}\_K^*\; \backslash to\; K^*\; \backslash to\; \backslash mathcal\{I\}\_K\; \backslash to\; \backslash mathcal\{C\}\_K\; \backslash to\; 0$

associated to every number field.

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

$I\; =\; (\backslash mathfrak\{p\}\_1\backslash ldots\backslash mathfrak\{p\}\_n)(\backslash mathfrak\{q\}\_1\backslash ldots\backslash mathfrak\{q\}\_m)^\{-1\}$

for

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

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

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

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

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Examples

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

(2\zeta_3 + 1)^2 &= 4\zeta_3^2 + 4\zeta_3 + 1 \\ &= 4(\zeta_3^2 + \zeta_3) + 1 \end{align}

Since $\backslash zeta\_3$ satisfies $\backslash zeta\_3^2\; +\; \backslash zeta\_3\; =-1$, our factorization makes sense.

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

: $I\; =\; \backslash left(2,\; \backslash frac12\backslash sqrt\{-23\}\; -\; \backslash frac12\backslash right)$ and $J=\backslash left(4,\backslash frac12\backslash sqrt\{-23\}\; +\; \backslash frac32\backslash right)$

to get the ideal

:$IJ=\backslash left(\backslash frac12\backslash sqrt\{-23\}+\backslash frac32\backslash right).$

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Divisorial ideal Let $\backslash tilde\; I$ denote the intersection of all principal fractional ideals containing a nonzero fractional ideal $I$.

Equivalently,

$\backslash tilde\; I\; =\; (R\; :\; (R\; :\; I)),$

where as above

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

If $\backslash tilde\; I\; =\; I$ 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 ring Krull domain (e.g., a Noetherian ring 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.

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See also

• Divisorial sheaf
• Dedekind-Kummer theorem

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Notes

• Chapter 9 of
• Chapter VII.1 of
• Chapter 11 of

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