Elementary equational proofs are supplied for Jacobson's theorem with fixed odd exponents and for Herstein's theorem with fixed exponents 4 and 8.
Brandenburg, Equational proofs of Jacobson’s theorem,arxiv:2310.05301
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Elementary proofs of ring commutativity theorems
Elementary equational proofs are supplied for Jacobson's theorem with fixed odd exponents and for Herstein's theorem with fixed exponents 4 and 8.