A classical theorem by Jacobson says that a ring in which every element x
satisfies the equation x^n = x for some n > 1 is commutative. According to Birkhoff’s Completeness Theorem, if n is fixed, there must be an equational proof of this theorem. The following paper makes a reduction to the case that n is a prime power p^k and the ring has characteristic p and prove the special cases k = 1 and k = 2. Try coming up with a proof for the case k = 3.
Reference paper: https://arxiv.org/pdf/2310.05301
If you are inspired by this idea, you can reach out to the authors for collaboration or cite it:
@misc{ren-equational-proofs-of-2026,
author = {Ren, Andrew},
title = {Equational Proofs of Jacobson's Commutativity Theorem when k = 3},
year = {2026},
url = {https://hypogenic.ai/ideahub/idea/hoQa3nuaM2gAaG5w2Zw5}
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