# A note on orders in quaternion algebras

Definition. An algebra $$B$$ over a field $$F$$ is a quaternion algebra if there exists $$i,j\in B$$ such that $$1,i,j,ij$$ is a basis for $$B$$ as a vector space over $$F$$, where $$i^2=a,j^2=b;a,b\in F^\times$$.

Throughout, we fix $$F=\mathbb{Q}$$.

Definition. A lattice in a finite-dimensional $$\mathbb{Q}$$-algebra $$V$$ is a finitely generated $$\mathbb{Z}$$-submodule $$\mathcal{L} \subset V$$ that contains a $$\mathbb{Q}$$-basis of $$V$$ (as a vector space over $$\mathbb{Q}$$).

I proved that

Let $$V_\mathbb{Q}$$ be a finite-dimensional vector space. $$\mathcal{L} \subset V$$ is a lattice if and only if $$\mathcal{L}=x_1\mathbb{Z} \oplus \ldots \oplus x_n \mathbb{Z}$$ where $$x_1,\ldots,x_n$$ is a basis for $$V_\mathbb{Q}$$.

Definition. An order $$\mathcal{O} \subset B$$ is a lattice that is also a subring having $$1\in \mathcal{O}$$.

I have two questions:

1. Are any two orders isomorphic as rings?!.
2. What are the orders in the quaternion algebra $$\mathbb{M}_2(\mathbb{Q})$$ (the matrix ring over $$\mathbb{Q}$$)?!. I found that the following subrings are all orders: \begin{align} &\begin{pmatrix} \mathbb{Z} &n\mathbb{Z} \\ m\mathbb{Z} & \mathbb{Z} \end{pmatrix}, \quad m,n\in \mathbb{Z}^* \\ &\begin{pmatrix} \mathbb{Z} & \frac{a}{d}\mathbb{Z} \\ n\mathbb{Z} & \mathbb{Z} \end{pmatrix}, \quad a,d,n\in \mathbb{Z}^* \text{ and } d \mid n, \\ &\begin{pmatrix} \mathbb{Z} & \frac{a}{b}\mathbb{Z} \\ \frac{c}{d}\mathbb{Z} & \mathbb{Z} \end{pmatrix}, \quad a,b,c,d\in \mathbb{Z}^* \text{ and } bd\mid ac. \end{align} Are there more orders?!. Observe that these three types of orders are all isomorphic as rings.

I appreciate any help. Thanks in advance.

• Might the relevant tag be integer-lattices or vector-lattices? It doesn't seem like 'lattice-order' is related here. Commented May 14, 2023 at 21:02