On tame and wild automorphisms of algebras

Let $B_n$ be a polynomial algebra of $n$ variables over a field $F$. Considering a free associative algebra $A_n$ of rank $n$ over $F$ as a polynomial algebra of noncommuting variables, we choose the ideal $R$ of all polynomials with a zero absolute term in $B_n$ and $A_n$. The well-known concept of wild automorphisms of the algebras $A_n$ and $B_n$ is transferred to $R$; the study of wild automorphisms is reduced to monic automorphisms of the algebra $R$, i.e., those identical on each factor $R^k/R^{k+1}$. In particular, this enables us to study the properties of the known Nagata and Anik automorphisms in detail. For $n=3$ we investigate the hypothesis that the Anik automorphism is tame modulo $R^k$ for every given integer $k>1$.