order in an algebra

Let $A$ be an algebra (not necessarily commutative), finitely generated over $\mathbb{Q}$. An order $R$ of $A$ is a subring of $A$ which is finitely generated as a $\mathbb{Z}$-module and which satisfies $R\otimes \mathbb{Q}=A$.

Examples:

1. The ring of integers in a number field is an order, known as the maximal order.

2. Let $K$ be a quadratic imaginary field and ${𝒪}_K$ its ring of integers. For each integer $n\ge 1$ the ring $𝒪=\mathbb{Z}+n{𝒪}_K$ 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 of the order $𝒪$.

Reference: Joseph H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, New York, 1986.

Title	order in an algebra
Canonical name	OrderInAnAlgebra
Date of creation	2013-03-22 13:41:22
Last modified on	2013-03-22 13:41:22
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	10
Author	alozano (2414)
Entry type	Definition
Classification	msc 06B10
Related topic	ComplexMultiplication
Defines	order
Defines	maximal order
Defines	conductor of an order