# fractional ideal

## 1 Basics

Let $A$ be an integral domain with field of fractions $K$. Then $K$ is an $A$–module, and we define a of $A$ to be a submodule of $K$ which is finitely generated as an $A$–module.

The product of two fractional ideals ${\mathfrak{a}}$ and ${\mathfrak{b}}$ of $A$ is defined to be the submodule of $K$ generated by all the products $x\cdot y\in K$, for $x\in{\mathfrak{a}}$ and $y\in{\mathfrak{b}}$. This product is denoted ${\mathfrak{a}}\cdot{\mathfrak{b}}$, and it is always a fractional ideal of $A$ as well. Note that, if $A$ itself is considered as a fractional ideal of $A$, then ${\mathfrak{a}}\cdot A={\mathfrak{a}}$. Accordingly, the set of fractional ideals is always a monoid under this product operation, with identity element $A$.

We say that a fractional ideal ${\mathfrak{a}}$ is invertible if there exists a fractional ideal ${\mathfrak{a}}^{\prime}$ such that ${\mathfrak{a}}\cdot{\mathfrak{a}}^{\prime}=A$. It can be shown that if ${\mathfrak{a}}$ is invertible, then its inverse must be ${\mathfrak{a}}^{\prime}=(A:{\mathfrak{a}})$, the annihilator11In general, for any fractional ideals ${\mathfrak{a}}$ and ${\mathfrak{b}}$, the annihilator of ${\mathfrak{b}}$ in ${\mathfrak{a}}$ is the fractional ideal $({\mathfrak{a}}:{\mathfrak{b}})$ consisting of all $x\in K$ such that $x\cdot{\mathfrak{b}}\subset{\mathfrak{a}}$. of ${\mathfrak{a}}$ in $A$.

## 2 Fractional ideals in Dedekind domains

We now suppose that $A$ is a Dedekind domain. In this case, every nonzero fractional ideal is invertible, and consequently the nonzero fractional ideals in $A$ form a group under ideal multiplication, called the ideal group of $A$.

The unique factorization of ideals theorem states that every fractional ideal in $A$ factors uniquely into a finite product of prime ideals of $A$ and their (fractional ideal) inverses. It follows that the ideal group of $A$ is freely generated as an abelian group by the nonzero prime ideals of $A$.

A fractional ideal of $A$ is said to be principal if it is generated as an $A$–module by a single element. The set of nonzero principal fractional ideals is a subgroup of the ideal group of $A$, and the quotient group of the ideal group of $A$ by the subgroup of principal fractional ideals is nothing other than the ideal class group of $A$.

Title fractional ideal FractionalIdeal 2013-03-22 12:42:38 2013-03-22 12:42:38 djao (24) djao (24) 5 djao (24) Definition msc 13A15 msc 13F05 IdealClassGroup ideal group