The results in the paper under review represent a further significant step towards a better understanding of the structure of PI algebras over a field of positive characteristic. An earlier result of the same author [Isr. J. Math. 81, No. 3, 343-355 (1993; Zbl 0795.16017)]shows that every PI algebra over a field of positive characteristic satisfies the complete linearization of the identity \(x^n=0\). In the present paper it is established that the connection between PI and nil algebras (of bounded nil index) is even stronger. Namely, it is shown that if \(A\) is a PI algebra there exists \(q\) with the following property. For every sufficiently large \(n\), \(A\) satisfies the partial linearizations of \(x^n=0\) having degrees \(\leq n/q\) in each variable. As a corollary of this theorem some properties of the radical of relatively free algebras are derived. It is shown that if \(F\) is a relatively free algebra of countable rank in a variety then any T-ideal in \(F\) generated by a subset of polynomials in the radical of \(F\), and depending on a finite number of variables, is nil of bounded index. This implies that the radical of a relatively free algebra in a nonmatrix variety must be nil of bounded index.
The results and the methods in the paper are of importance for the theory of algebras with identities over a field of positive characteristic. This theory is rather new, and the difference between it and its counterpart in characteristic 0 is significant. One of the main difficulties in the case of positive characteristic is that in general one cannot linearize / symmetrize a given polynomial; and thus the multilinear identities in an algebra do not determine its T-ideal. The paper under review makes a step towards resolving this problem; or at least showing to what extent one can symmetrize a polynomial without loss of “equivalence” of the consequences.
For the entire collection see [Zbl 0879.00041].