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.