# fractional ideal

## 1 Basics

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 annihilator  11In 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 $A$.

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