The open case for simple alternative rings

Abstract

2. Shirshov's result. We now state a convenient version of one of Shirshov's theorems. Let X be any set (of 'indeterminates'). Let Nx(F) and Sx(F) be, respectively, the free nonassociative and free associative algebras over the field F and on the free generating set X. There is an obvious F-homomorphism a from Nx(F) onto Sx(F): in the standard representation az deletes all parentheses. Say p CNx(F) is admissible provided pa;50. Now suppose R is any (nonassociative) algebra over F. Say R satisfies the admissible polynomial identity (p.i.) p, provided p ENx(F) is admissible, and for every F-homomorphism 3 of Nx(F) into R, po3=O. We canl now state