Generalized Quaternion Rings over $$\mathbb {Z}/n\mathbb {Z}$$ Z / n Z for an Odd $$\varvec{n}$$ n

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 R-isomorphic generalized quaternion rings \(\Big (\frac{a,b}{\mathbb {Z}/n\mathbb {Z}}\Big )\).