A Few Words Concerning Affine PI-Algebras: Shirshov’s Theorem
Affine algebras were defined in Definition 1.1.8. Perhaps the first major triumph of PI-theory was Kaplansky’s proof [Kap50] of A.G. Kurosh’s conjecture for PI-algebras, that every algebraic affine PI-algebra over a field is finite dimensional. (Later a non-PI counterexample to Kurosh’s conjecture was discovered by Golod and Shafarevich [Gol64].) Kaplansky’s proof was structural, based on Jacobson’s radical and Levitzki’s locally finite radical.