FACTOR ALGEBRAS OF FREE ALGEBRAS: ON A PROBLEM OF G. BERGMAN

Abstract

Let $A_n=K\langle x_1,\ldots,x_n \rangle$ be a free associative algebra over a field $K$. In this paper, examples are given of elements $u \in A_n$, $n \ge 3$, such that the factor algebra of $A_n$ over the ideal generated by $u$ is isomorphic to $A_{n-1}$, and yet $u$ is not a primitive element of $A_n$ (that is, it cannot be taken to $x_1$ by an automorphism of $A_n$). 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.