I'm having a hard time finding information about the quotient rings of the Lipschitz quaternions and the Hurwitz quaternions.

The Lipschitz quaternions are defined as the quaternions with integral coordinates. The Hurwitz quaternions are those with all coordinates in $\mathbb Z$ or in $\mathbb Z + \frac{1}{2}$. It is quite well-known that the former lacks unique factorization in irreducible elements, while the latter admits a Euclidean division with respect to the norm $N(a+ib+jc+kd) = a^2+b^2+c^2+d^2$.

I'm interested in the quotient rings of these rings. Is it, in principle, possible to determine the quotient of the (left)-ideal generated by (say) $a+ib+jc+kd$? My guess would be that this ideal has index $a^2+b^2+c^2+d^2$ in the corresponding ring, based on the analogous result in the Gaussian integers. And what about two-sided ideals?

My motivation is to realize some given $\mathbb Z/n\mathbb Z$ as such a quotient, but I'm afraid I can only get noncommutative quotients.