GAAG seminar - On an old theorem of Jacobson

Abstract: It is well-known and easy to prove that any ring in which every element equals its own square is commutative. Proving the same thing with "square" replaced by "cube" is already more complicated, and the difficulties increase with higher powers. Using his theory of algebras with vanishing radicals, Jacobson proved in 1945 that every ring in which each element equals some power of itself, with an exponent that is greater than 1 but that may depend on the element, is necessarily commutative. The lecture is devoted to a novel proof that, other than existing proofs, has attractive algorithmic features. In particular, it gives rise to a method that, when the exponent in Jacobson's theorem is fixed and given, constructs a perfectly elementary proof that depends on no more than ring-theoretic identities. The lecture is based on joint work with Mike Daas, Lars Pos, and Samuel Tiersma.