### 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 $s \in S$ such that whenever $g(x_1^n) \in I$ for all $x_1^n \in R$, then $g(s,x_i,1^{(n-2)}) \in I$ for some $1 \leq i \leq n$. 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.

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