On the nilindex of the radical of a relatively free associative algebra.
A classical result of Amitsur states that the Jacobson radical of a relatively free associative algebra over an infinite field is nil; Kemer showed that each finitely generated T-ideal contained in the Jacobson radical of a relatively free algebra is nil of bounded index. Let \(F\) be an infinite field of positive characteristic \(p\) and let \(F(X)\) be the free associative algebra on an infinite set of variables \(X\). If \(\Gamma\) is a T-ideal in \(F(X)\) then the complexity of \(\Gamma\) equals the largest positive integer \(n\) such that the T-ideal of the full matrix algebra \(M_n(F)\) contains \(\Gamma\). The main theorem of the paper under review states that if the complexity \(n\) of \(\Gamma\) satisfies \(n<p\) then the Jacobson radical of \(F(X)/\Gamma\) is nil of bounded index. In the course of proof the author proves the following result that is quite interesting. If \(n<p\) then every trace identity for \(M_n(F)\) follows from the identitites \(T(1)=n\) and from the complete linearization of the Cayley-Hamilton identity. Here \(T(a)\) stands for the trace of the matrix \(a\).
scientific article; zbMATH DE number 5319866