"Is it true that any division ring is a domain?"
Note 1: I am not sure "domain"="integer domain", are they different?
Note 2: Since the definition of integral domain, I can't see if a division ring MUST be commutative, the nonzero elements form a group under multiplication may not be abelian.
So, how can I prove that "Any division ring is a domain" ?
Best Answer
If domain means integral domain, then division rings need not be domains because integral domains are commutative. It is conceivable that an unqualified "domain" could mean "a ring with no zero divisors." If that is what the author is using it to mean, then the statement is true: division rings cannot have zero divisors because every nonzero element is invertible.
For a proof, suppose $ab=0$ and $a\neq 0$. Then $$a^{-1}(ab)=b=0=a^{-1}0$$ Thus whenever $ab=0$, either $a=0$ or $b=0$.