On Hyperideals of Multiplicative Hyperrings

Abstract

Let R be a commutative multiplicative hyperring. In this paper, we introduce and study the concepts of n-hyperideal and δ-n-hyperideal of R which are generalization of n-ideals and δ-n-ideals of the in a commutative ring. An element a is called a nilpotent element of R if there exists a positive integer n such that 0∈a^n. A hyperideal I (I ≠R) of R is called an n- hyperideal of R if for all a,b∈R, a*b⊆I and a is non-nilpotent element implies that b∈I [15]. Also, I is called a δ-n-hyperideal if for all a,b∈R, a*b⊆I then either a is nilpotent or b∈δ(I) , where δ is an expansion function over the set of all hyperideals of a multiplicative hyperring. In addition, we give the definition of zd-hyperideal. Some properties of n-hyperideals, δ-n-hyperideals and zd-hyperideals of the hyperring R are presented. Finally, the relations between these notions are investigated.