fractional ideal FractionalIdeal 2002-06-02 00:15:16 2002-06-02 00:20:32 Definition ideal group % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \renewcommand{\a}{{\mathfrak{a}}} \renewcommand{\b}{{\mathfrak{b}}} \section{Basics} Let $A$ be an integral domain with field of fractions $K$. Then $K$ is an $A$--module, and we define a {\em 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 {\em 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\footnote{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 $x\cdot\b \subset \a$.} of $\a$ in $A$. \section{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 {\em ideal group} of $A$. The {\em 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 {\em 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 $A$.