What is an example of integral domain that is not a division ring?
Integral domain that is not a division ring
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$\begingroup$
abstract-algebra
ring-theory
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5Have you tried anything? What integral domains do you know? – 2011-09-30
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2Please ask a question. – 2011-09-30
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2ZZZZZzzzzzzzzzzzzzzzz.... – 2011-09-30
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0@Lmn6 Not a well posed question. – 2011-09-30
3 Answers
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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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0...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...) – 2011-09-30
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0Swlabr: integral numbers... (use * ... *) – 2011-09-30
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0Ah, I've never come across them called that before. Thanks. (That doesn't seem to work - it just gives *l* or $*l*$...) – 2011-09-30
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1I once went to a lecture titled "integral equations" expecting to hear about equations involving integrals. Instead I heard about equations with coefficients in $\mathbb Z$. – 2011-09-30
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0Definitions 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. – 2011-09-30
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0@Chris: was that intended for me or Swlabr? If for Swlabr, using $\mathrm{@Swlabr}$ notifies Swlabr. – 2011-09-30
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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.