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Let R be a Krasner (m,n)-hyperring and S be an n-ary multiplicative subset ofR. The purpose of this paper is to introduce the notion of n-ary S-primehyperideals as a new expansion of n-ary prime hyperideals. Several propertiesand characterizations concerning n-ary S-prime hyperideals are presented. Thestability of this new concept with respect to various hyperring-theoreticconstructions are studied. Furthermore, we extend this concept to n-aryS-primary hyperideals. We obtained some specific results explaining thestructure.

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