The Complemented Semiring And Injective Semimodules

In this paper we will continue to discuss the relationship between ζ_s -injective modules and the exact sequences on the basis of paper [13] and [19]; The ζ_s-injective dimension is introduced to classify preliminary semirings. It is proved that a semiring S has ζ_s -injective modules if and only if S is a non-zeroic semirings. In addition, the complemented semiring is defined and it is proved that the complemented semiring is also non-zeroic semiring and so there exist be non—zero ζ_s-injective modules. It is proved that a semiring is complemented if and only if it is a direct product of some Boolean algebra and Boolean ring by studying the relations of congruences on the complemented semiring, so the complemented semiring is the commutative semiring. Finally, it is proved that any semimodule over complemented semirings must have a ζ_s-injective hull and the complemented semiring must be the rPP semiring and the endomorphism semiring of every principal right ideal is still complemented semiring.