Factor algebras of free algebras: On a problem of G. Bergman

Abstract

Let An = K〈X1,...,Xn〉 be a free associative algebra over a field K. In this paper, examples are given of elements u ∈ An, n ≥ 3, such that the factor algebra of An over the ideal generated by u is isomorphic to An-1, and yet u is not a primitive element of An (that is, it cannot be taken to x1 by an automorphism of An). If the characteristic of the ground field K is 0, this yields a negative answer to a question of G. Bergman. On the other hand, by a result of Drensky and Yu, there is no such example for n = 2. It should be noted that a similar question for polynomial algebras, known as the embedding conjecture of Abhyankar and Sathaye, is a major open problem in affine algebraic geometry.