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I have this problem:

Prove that if $R$ is a ring such that all its subrings are division rings, then $R$ must be a field.

The problem is that the suggestion is to use the Periocidity theorem of Jacobson, but I don't find it by that name. Can anyone say me which theorem could be the one that is reffered as the suggestion and in which book find it?

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Perhaps it is this one:

If $R$ is a ring such that for every $x\in R$ there exists a natural number $n$ such that $x^n=x$, then $R$ is commutative.

That certainly sounds like periodicity, and moreover it is a commutativity result, which would explain why your division ring is a field (if you can tell how the periodicity theorem applies, here.)

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  • $\begingroup$ Thanks, I will try the problem with this. $\endgroup$ – MonsieurGalois Dec 16 '16 at 20:41
  • $\begingroup$ @MonsieurGalois Here's a relevant post on that theorem. $\endgroup$ – rschwieb Dec 16 '16 at 20:44

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