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"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" ?

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  • $\begingroup$ "integral domains" are (nearly) always assumed to be commutative. There are authors who use just "domain" to apply to noncommutative rings without nonzero zero divisors. $\endgroup$ – rschwieb Jan 4 '15 at 11:55
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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 $b=0$.

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  • $\begingroup$ "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." Would you like explain more? Why invertible can imply no zero divisors? $\endgroup$ – Richard Jan 4 '15 at 11:35
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    $\begingroup$ @user131605 please see edit. $\endgroup$ – Matt Samuel Jan 4 '15 at 11:38
  • $\begingroup$ @user131605 you're welcome. $\endgroup$ – Matt Samuel Jan 4 '15 at 11:45
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Assume every element has an inverse (i.e. the ring is a division ring) and show that no two elements multiply to zero - then it must also be an integral domain.

Suppose $ab =0$. Either neither element is zero, or at least one is zero. In the first case since we are in a division ring, we can multiply each side by the inverse of $a$. This gives $b=0$. In the second case we already know one of $a$ or $b$ is zero.

This shows there are no zero divisors in a division ring.

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The quaternions are a division ring. They are non commutative. These types of rings are sometimes called non-commutative domains.

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