  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 $n \not
\equiv 0 \pmod{4}$.
