$n$ is less than $\Upsilon (n, l),$ where $$\Upsilon (n, l) = 2n^{3\ulcorner\log_3 n\urcorner + 4} l.$$

In [@Bog01] it is established that the nilpotency index of an $l$-generated nil-semiring of degree $n$ equals the nilpotency index of an $l$-generated nilring of degree $n$, where addition is not supposed to be commutative. (The paper also contains examples of non-nilpotent nil-nearrings of index $2$.) Thus our results extend to the case of semirings as well.

On estimates from below
-----------------------

Let us compare the results obtained with the estimate for height from below. The height of an algebra $A$ is not less than its Gelfand — Kirillov dimension $\GK(A)$. For the algebra of $l$-generated general matrices of order $n$ this dimension equals $(l-1)n^2+1$ (see [@Procesi] as well as [@Bel04]). At the same time, the minimal degree of an identity in this algebra is $2n$ by Amitsur — Levitsky theorem. We have

The height of an $l$-generated $\PI$-algebra of degree $n$ and of the set of not $n$-divisible words over an alphabet of cardinality $l$ is not less than $(l-1)n^2/4+1$.

Estimates from below for the nilpotency index were established by Ye. N. Kuzmin in [@Kuz75]. He gave an example of $2$-generated algebra 