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

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