Gröbner–Shirshov bases for semirings

In the paper we derive a Gröbner–Shirshov algorithm for semirings and commutative semirings. As applications, we obtain Gröbner–Shirshov bases and A. Blassʼs (1995) and M. Fiore and T. Leinsterʼs (2004) normal forms of the semirings N[x]/(x=1+x+x2) and N[x]/(x=1+x2), correspondingly.