The Nagata–Higman theorem for semirings

This paper contains the proof of the Nagata–Higman theorem for semirings (with non-commutative addition in general). The main results are the following:
Theorem. Let $A$ be an $l$-generated semiring with commutative addition in which the identity $x^{m}=0$ is satisfied. Then the nilpotency index of $A$ is not greater than $2l^{m+1}m^{3}$.
Nagata–Higman theorem for general semirings. If an $l$-generated semiring satisfies the identity $x^{m}=0$ than every word in it of length greater than $m^{m}\cdot2l^{m+1}m^{3}+ m$ is zero.