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 ${\mathcal{O}}_{K}$ its ring of integers. For each integer $n\ge 1$ 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 of the order $\mathcal{O}$.
Reference: Joseph H. Silverman, The arithmetic of elliptic curves, SpringerVerlag, New York, 1986.
Title  order in an algebra 

Canonical name  OrderInAnAlgebra 
Date of creation  20130322 13:41:22 
Last modified on  20130322 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 