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

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 1(xn1) ? I for all xn1 ? R, then 1(s,xi,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, the concept of n-ary S-primary hyperideals is introduced. Several properties of them are provided.