Ideal theory in Γ -semihyperrings

The concept of Γ -semihyperring is a generalization of semiring, a generalization of semihyperring and a generalization of Γ -semiring. Since the theory of ideals plays an important role in the theory of Γ - semihyperring, in this paper, we will make an intensive study of the notions of Noetherian, Artinian, simple and regular Γ -semihyperrings. The bulk of this paper is devoted to stating and proving analogues to several theorems in the theory of Γ -semihyperrings.