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?


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.)

  • $\begingroup$ 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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