with identity $x^n=0$, such that its nilpotency index exceeds $(n^2+n-2)/2$. The problem of finding estimates from below is considered in [@Kh11].

At the same time, for zero characteristic and a countable set of generators, Yu. P. Razmyslov (see for instance [@Razmyslov3]) obtained an upper estimate for the nilpotency index, namely $n^2$.

First we will prove theorem \[c:main1\], and in the following section we will deal with estimates for essential height, that is, for the number of distinct periodical pieces in a not $n$-divisible word.

[**Acknowledgements.**]{} The authors are grateful to V. N. Latyshev, A. V. Mikhalyov and all participants of the seminar “Ring theory” for their attention to our work, as well as to the participants of the seminar at Moscow Physic-technical institute under the guidance of A. M. Raigorodsky.

Estimates on occurence of degrees of subwords
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The outline of the proof for theorem \[c:main1\]
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Lemmas \[Lm0.1\], \[c:lem1.2\] and \[c:lem1.3\] describe sufficient conditions for presence of a period of length $d$ in a not $n$-divisible word $W$. Lemma \[c:lem1.4\] connects $n$-divisibility of a word $W$ with the set of its tails. Further we choose some specific subset in the set of tails of $W$, such that we can 