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What is an example of integral domain that is not a division ring?

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Have you tried anything? What integral domains do you know? – Chris Eagle Sep 30 '11 at 13:20
Please ask a question. – Rasmus Sep 30 '11 at 13:20
ZZZZZzzzzzzzzzzzzzzzz.... – shaye Sep 30 '11 at 13:38
@Lmn6 Not a well posed question. – user38268 Sep 30 '11 at 13:49
up vote 6 down vote accepted

What is the first ring that pops into your head when you think of a ring? Is it a division ring? No! Is it an integral domain? Yes!

Methinks you just need to learn your definitions...

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Hint: think about the word integral

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...I'm not getting this hint. I mean...$L^p$-space? But that's a vector space. You can't really multiply integrals, can you? (Also, how can you do script letters here? \mathscr doesn't seem to work...) – user1729 Sep 30 '11 at 13:41
Swlabr: integral numbers... (use * ... *) – Rasmus Sep 30 '11 at 13:48
Ah, I've never come across them called that before. Thanks. (That doesn't seem to work - it just gives l or $*l*$...) – user1729 Sep 30 '11 at 13:54
I once went to a lecture titled "integral equations" expecting to hear about equations involving integrals. Instead I heard about equations with coefficients in $\mathbb Z$. – GEdgar Sep 30 '11 at 14:35
Definitions are so important. The "division" in division ring does not mean that the ring has a division algorithm (i.e., an analogue of the Euclidean algorithm). Always check definitions if you are not sure what the term means. Any decent abstract algebra text will answer your question. If none is handy, try Google. – Chris Leary Sep 30 '11 at 14:54

In a division ring every non zero element is a unit. The only units in $\mathbb{Z}$ are $1$ and $-1$, so $\mathbb{Z}$ is not a division ring but an integral domain.

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