On the nilindex of the radical of a relatively free associative algebra.

In the paper, it is proved that the radical of a relatively free associative algebra of countable rank over an infinite field of characteristic p > 0 is a nil ideal of bounded index if the complexity of the corresponding variety is less than p. Moreover, a description of a basis for trace identities for the matrix algebra M _n over an infinite field of characteristic p > 0, n < p, is obtained in the paper. [ABSTRACT FROM AUTHOR]