# 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. 1.

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

2. 2.

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 of the order $\mathcal{O}$.

Reference: Joseph H. Silverman, , Springer-Verlag, New York, 1986.

Title order in an algebra OrderInAnAlgebra 2013-03-22 13:41:22 2013-03-22 13:41:22 alozano (2414) alozano (2414) 10 alozano (2414) Definition msc 06B10 ComplexMultiplication order maximal order conductor of an order