On the structure of quaternion rings over $\mathbb{Z}/n \mathbb{Z}$

In this paper we investigate the structure of $\left(\frac{a,b}{\Z{n}}\right)$, the quaternion rings over $\Z{n}$. It is proved that these rings are isomorphic to $\left(\frac{-1,-1}{\Z{n}}\right)$ if $a \equiv b\equiv -1 \pmod{4}$ or to $\left(\frac{1,1}{\Z{n}}\right)$ otherwise. We also prove that the ring $\left(\frac{a,b}{\Z{n}}\right)$ is isomorphic to $\mathbb{M}_2(\Z{n})$ if and only if $n$ is odd and that all quaternion algebras defined over $\Z{n}$ are isomorphic if and only if $n \not \equiv 0 \pmod{4}$.