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 $\a$ and $\b$ of $A$ is defined to be the submodule of $K$ generated by all the products $x \cdot y \in K$ , for $x \in \a$ and $y \in \b$ . This product is denoted $\a \cdot \b$ , 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 $\a \cdot A = \a$ . Accordingly, the set of fractional ideals is always a monoid under this product operation, with identity element $A$ .

We say that a fractional ideal $\a$ is invertible if there exists a fractional ideal $\a'$ such that $\a \cdot \a' = A$ . It can be shown that if $\a$ is invertible, then its inverse must be $\a' = (A:\a)$ , the annihilator ¹ of $\a$ in $A$ .

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 $A$ .

Footnotes

...http://planetmath.org/encyclopedia/Annihilator4.html ¹

In general, for any fractional ideals $\a$ and $\b$ , the annihilator of $\b$ in $\a$ is the fractional ideal $(\a:\b)$ consisting of all $x \in K$ such that $x\cdot\b \subset \a$ .