# Generalized Quaternion Rings over $\mathbb{Z}/n\mathbb{Z}$ for an odd $n$

## TL;DR

This paper explores a broad class of quaternion rings over the integers modulo an odd number, providing a classification and counting the distinct isomorphism types for these generalized structures.

## Contribution

It introduces a generalization of quaternion rings over rac{a,b}{R} that includes non-unit elements and computes the number of non-isomorphic rings over rac{a,b}{\u00bb} for odd n.

## Key findings

- Derived formulas for counting non-isomorphic quaternion rings
- Extended the theory to include non-unit elements in quaternion rings
- Provided explicit classifications for odd modulus cases

## Abstract

We consider a generalization of the quaternion ring $\Big(\frac{a,b}{R}\Big)$ over a commutative unital ring $R$ that includes the case when $a$ and $b$ are not units of $R$. In this paper, we focus on the case $R=\mathbb{Z}/n\mathbb{Z}$ for and odd $n$. In particular, for every odd integer $n$ we compute the number of non-isomorphic generalized quaternion rings $\Big(\frac{a,b}{\mathbb{Z}/n\mathbb{Z}}\Big)$