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 $n\geq 1$ 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 conductor of the order $\mathcal{O}$ .

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