ABSTRACT
Affine algebras were defined in Definition 1.74. Although much of the structure theory of affine Pi-algebras has been well understood for some time and exposed in [KrLeOO] and [Row88, Chapter 6.3], many of these results can be streamlined and improved when viewed combinatorically in terms of words. 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 Kurosh's conjecture was disproved in general by Golod and Shafarevich [Gol64].) Kaplansky's proof was structural, based on Jacobson's radical and Levitzki's locally finite radical. The first combinatorial proof, obtained by A.I. Shirshov, was seen later to be a consequence of Shirshov's Height Theorem, which does not require any assumption on algebraicity, and shows that any affine Pi-algebra bears a certain resemblance to a commutative polynomial algebra, thereby yielding an immediate solution to Kurosh's conjecture for Pi-algebras. Another consequence of Shirshov's Height Theorem, proved by Berele, is the finiteness of the Gelfand-Kirillov dimension of any affine Pi-algebra. A crucial dividend of Shirshov's approach is that it generalizes at once to an affine algebra over an arbitrary commutative ring:
If A = C { a i , . . . ,a¿} is an affine Pi-algebra over a commutative ring C and a certain finite set of elements of A (e.g., the words in the a¿ of degree up to some given number to be specified) is integral over C, then A is f.g. (finitely generated) as a C-module.
