My professor put this problem in a list of $200$ problems for the examen:
A ring $R$ is called periodic if for each $x\in R$ exist $n\geq 2$ such that $x^n=x$. Starting from the Jacobson Theorem (All periodic rings are commutative) conclude that:
- Any finite division ring is a field.
- If $R$ is a ring such that any subring is a division ring, then $R$ is a field.
The first one is the Wedderburn theorem, and I found proofs in books and in internet, but the second one I don't know how to proceed with it. I would appreciate if anyone can give me a hint or a reference to read the proof.