Fully idempotent semirings
Ahsan Javed · Proceedings of the Japan Academy Series A Mathematical Sciences · 1993
In this paper A will denote a semiring (A, -+-, ") as defined, for exam- ple, in [2], that is, two semigroups (A, +) and (A, ") such that addition dis- tributes over multiplication.Moreover, we shall always assume that (A, +)is commutative and (A, +, ")has an absorbing zero 0, that is, a+ 0--0 +a aand0"a-" a'0 0 hold for allaA.The notions of left, right, and two-sided ideals, as well as sums and products of such ideals are defined as usual.The word ideal will always mean a two-sided ideal.An ideal P of A is called prime (irreducible; strongly irreducible) if IJ -P I -P or J-P(IJ=PI=P or J=P;ICJ_PI_P orJ -P)