06B10-OrderInAnAlgebra.tex

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\pmtitle{order in an algebra}
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\pmtype{Definition}
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%\pmkeywords{maximal order}
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\pmdefines{conductor of an order}

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\begin{document}
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.
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