Abstract
In this paper we study the isomorphisms of generalized Hamilton quaternions $\Big(\frac{a,b}{R}\Big)$ where $R$ is a finite unital commutative ring of odd characteristic and $a,b \in R$. We obtain the number of non-isomorphic classes of generalized Hamilton quaternions in the case where $R$ is a principal ideal ring. This extends the case $R=\mathbb{Z}/n\mathbb{Z}$
 where $n$ is an odd integer.
Highlights
The origin of quaternions dates back to 1843, when William Rowan Hamilton considered a 4-dimensional vector space over R with basis {1, i, j, k} and defined an associative product given by the classical rules i2 = j2 = −1 and ij = −ji = k.This construction admits a very natural extension like the following
In this paper we study the isomorphism classes of generalized Hamilton quaternions a,b R
We can provide the required extension of Lemma 2.1 to finite local rings that will be used in the sequel
Summary
The origin of quaternions dates back to 1843, when William Rowan Hamilton considered a 4-dimensional vector space over R with basis {1, i, j, k} and defined an associative product given by the classical rules i2 = j2 = −1 and ij = −ji = k. This construction admits a very natural extension like the following. We obtain an associative unital ring that will be denoted by a,b R and that we call ring of generalized Hamilton quaternions over R If both a and b are units and the underlying ring R is a field F of characteristic different from 2, the structure of the corresponding quaternion algebra is well-known.
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