Geodesic
The Nagata--Higman theorem for semirings
Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 2, pp. 523-527.

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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. 
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A. Ya. Belov. The Nagata--Higman theorem for semirings. Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 2, pp. 523-527. http://geodesic.mathdoc.fr/item/FPM_1995_1_2_a11/

[1] A. J. Belov, “Some estimations for nilpotence of nill-algebras over a field of an arbitrary characteristic and height theorem”, Comm. in Algebra, 1992, 2919–2922 | DOI | MR | Zbl