TITLE: Are there domains that are not commutative?
QUESTION [5 upvotes]: Let $A$ a domain, i.e. $ab\in A\implies a=0$ or $b=0$. It's written that all domains are commutative. Is it by definition, or can we prove that domains are commutative? I mean, do we only consider domaind for commutative rings, or is a ring that is a domain then commutative?

REPLY [6 votes]: Some authors do allow "domain" to refer to a noncommutative ring. But nearly universally "integral domain" will imply commutativity.
Here are nine examples of noncommutative rings with no nonzero zero divisors.
I'll call out at least two of them, the Hurtwitz and Lipschitz quaternions.