Extensions of $n$-ary prime hyperideals via an $n$-ary multiplicative subset in a Krasner $(m,n)$-hyperring

Abstract

Let R be a Krasner (m, n)-hyperring and S be an n-ary multiplicative subset of R. The purpose of this paper is to introduce the notion of n-ary S-prime hyperideals as a new expansion of n-ary prime hyperideals. A hyperideal I of R disjoint with S is said to be an n-ary S-prime hyperideal if there exists s ∈ S such that whenever g(x n 1 ) ∈ I for all x n 1 ∈ R, then g(s, x i , 1 (n−2) ) ∈ I for some 1 ≤ i ≤ n. Several properties and characterizations concerning n-ary S-prime hyperideals are presented. The stability of this new concept with respect to various hyperring-theoretic constructions are studied. Furthermore, we extend this concept to n-ary S-primary hyperideals. We obtained some specific results explaining the structure.2010 Mathematics Subject Classification. 16Y99. Key words and phrases.n-ary J-hyperideal, n-ary δ-J-hyperideal, (k, n)-absorbing δ-Jhyperideal.