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 fractional ideal 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
Canonical name	FractionalIdeal
Date of creation	2013-03-22 12:42:38
Last modified on	2013-03-22 12:42:38
Owner	djao (24)
Last modified by	djao (24)
Numerical id	5
Author	djao (24)
Entry type	Definition
Classification	msc 13A15
Classification	msc 13F05
Related topic	IdealClassGroup
Defines	ideal group