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.



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