There are two notions of integer quaternions. One is the natural ring $L=\mathbb{H}(\mathbb{Z})$ comprising quaternions of the form $a+bi+cj+dk$ with $a,b,c,d \in \mathbb{Z}$ (the Lipschitz quaternions). A less obvious, but apparently more useful, notion is that of Hurwitz quaternions $H$ comprising elements of the form $a+bi+cj+dk$ where $$a,b,c,d \in \mathbb{Z} $$ or $$a,b,c,d \in \mathbb{Z} + \frac{1}{2}$$ One advantage in the Hurwitz quaternion ring $H$ is that there exists a (non-commutative) Euclidean division algorithm, and every left (or right) ideal is principal.

In the case of Lipschitz quaternions $L$, not every left (or right) ideal is principal, and this is mentioned in an answer to this question: Ideal class "group" of Lipschitz (fully-integer) quaternions The left (or right) ideal $(2, 1+i+j+k)$ is not principal, and the answer further shows that every ideal of $L$ is either principal, or of the form $(\alpha, \alpha.\frac{1+i+j+k}{2})$ for some $\alpha \in L$.

My question is regarding two-sided ideals of $L$ and $H$.

Is there a characterization of the two-sided ideals of $L$ and $H$? Which principal left ideals of $H$ are also right ideals?

My second question is regarding the quotients formed from the ideals.

Is every quotient ring (of $H$ and of $L$) formed using a non-trivial ideal, a finite ring? Is the structure of these rings known? Is there some characterization of the quotient rings?

For instance, $L$ modulo the ideal ($p$) for a prime $p$ gives us the ring $\mathbb{H}(\mathbb{F}_p)$ of quaternions over the finite field $\mathbb{F}_p$. But this is immediate since $p$ is in the center of $L$. I find it harder to work in more generality when the ideal is generated by an element not in the center. Or worse, when the ideal is not even principal.

Any directions or references are welcome too. I am just interested in the possible finite rings that arise as quotients of $L$ and $H$.

  • $\begingroup$ I had what you say in mind, but clearly thats not how I wrote it. Sorry and thanks, I have edited the question. $\endgroup$ – BharatRam Sep 20 '17 at 17:39

For the Hurwitz quaternions $H$, the non-zero two sided ideals are just $nH$ for $n\in\Bbb N$ and $nJ$ where $J$ consists of the elements of even norm in $H$.

To see this, note that any 2-sided ideal of $H$ contains some $mH$, and so corresponds to an ideal of $H/nH$. This is a direct sum of rings $H/p^r H$ with $p$ prime. When $p$ is odd, $H/p^r H$ is a polynomial ring over $\Bbb Z/p^r\Bbb Z$. The only two-sided ideals of this are generated powers of $p$.

The case $p=2$ is a bit more interesting. In this case $H/2^r H$ is a quotient of the valuation ring of the dimension four division algebra over the $2$-adic field. This has a maximal ideal $J$ with index $4$ whose square is generated by $2$.

For $L$ one has to think about what happens locally at the prime $2$. I haven't the time to consider this right now...

  • $\begingroup$ I can't remember, but I think Conway and Smith tell you the answer. $\endgroup$ – Kimball Sep 21 '17 at 23:58

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