order in an algebra OrderInAnAlgebra 2003-06-14 18:12:38 2007-05-15 12:19:57 Definition order maximal order conductor of an order order maximal order algebra ring of integers % 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 Let $A$ be an algebra (not necessarily commutative), finitely generated over $\mathbb{Q}$. An {\it order} $R$ of $A$ is a subring of $A$ which is finitely generated as a $\mathbb{Z}$-module and which satisfies $R \otimes \mathbb{Q}= A$. {\bf Examples:} \begin{enumerate} \item The ring of integers in a number field is an order, known as the {\it maximal order}. \item Let $K$ be a quadratic imaginary field and $\mathcal{O}_K$ its ring of integers. For each integer $n\geq 1$ the ring $\mathcal{O}={\mathbb{Z}}+n\mathcal{O}_K$ is an order of $K$ (in fact it can be proved that every order of $K$ is of this form). The number $n$ is called the {\it \PMlinkescapetext{conductor}} of the order $\mathcal{O}$. \end{enumerate} {\it Reference}: Joseph H. Silverman, {\it The arithmetic of elliptic curves}, Springer-Verlag, New York, 1986.