I have to give an example of a non-commutative domain that is not a division ring. My first thought was $R = \big\{ a + bi + cj + dk \mid a,b,c,d \in \mathbb{Z} \big\}$ since $R$ is clearly non-commutative and not a division ring (take the number 2 for example). I have seen conflicting accounts for if it is a domain though. One source said it is since it is a subring of the quaternions. However my book does not specify that quaternions are a domain, and I read a paper online that said that in fact they weren't a domain. Then I read an answer on stack exchange from someone with a 62k rep that said the quaternions are a domain. I tried proving it myself but I got stuck in a very long cycle.
2026-04-02 06:40:52.1775112052
Are the quaternions a domain?
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I think your confusion is coming from the fact that different authors use the word "domain" with different meanings. In the context of your problem, "domain" clearly just means a (nonzero) ring in which $xy=0$ implies $x=0$ or $y=0$. With that definition, any division ring is a domain (if $x\neq 0$ you can multiply $xy=0$ on the left by $x^{-1}$ to get that $y=0$), and since a subring of a domain is clearly a domain, it follows that $R$ is a domain.
On the other hand, some authors (especially those that think mainly about commutative rings) take "domain" to mean that in addition to the condition above, the ring is commutative. With this definition, of course the quaternions are not a domain, because they are not commutative.