Geodesic
The Nagata--Higman theorem for hemirings
Fundamentalʹnaâ i prikladnaâ matematika, Tome 7 (2001) no. 3, pp. 651-658.

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In this paper the hemirings (in general, with noncommutative addition) with the identity $x^n=0$ are studied. The main results are the following ones. Theorem. If a $n!$-torsionfree general hemiring satisfies the identity $x^n=0$, then it is nilpotent. The estimates of the nilpotency index are equal for $n!$-torsionless rings and general hemirings. Theorem. The estimates of the nilpotency index of $l$-generated rings and general hemirings with identity $x^n=0$ are equal. The proof is based on the following lemma. Lemma. If a general semiring $S$ satisfies the identity $x^n=0$, then $S^n$ is a ring. 
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     author = {I. I. Bogdanov},
     title = {The {Nagata--Higman} theorem for hemirings},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {651--658},
     publisher = {mathdoc},
     volume = {7},
     number = {3},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2001_7_3_a1/}
}
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I. I. Bogdanov. The Nagata--Higman theorem for hemirings. Fundamentalʹnaâ i prikladnaâ matematika, Tome 7 (2001) no. 3, pp. 651-658. http://geodesic.mathdoc.fr/item/FPM_2001_7_3_a1/