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

TL;DR

This paper characterizes the structure of quaternion rings over rac{n}{ } and determines conditions under which they are isomorphic to matrix rings or other quaternion rings, revealing their algebraic classifications.

Contribution

It provides a complete classification of quaternion rings over rac{n}{ }, including isomorphism conditions and structural equivalences, extending understanding of these algebraic objects.

Findings

Quaternion rings are isomorphic to rac{-1,-1}{ } or rac{1,1}{ } depending on congruence conditions.

Quaternion rings over rac{n}{ } are matrix rings rac{2}{ } if and only if n is odd.

All quaternion algebras over rac{n}{ } are isomorphic if and only if n rac{ ot ext{ divisible by 4}}{ }.

Abstract

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