Generalizations of Prime Hyperideals via Hypersystems in Krasner Hyperrings

The aim of this study is to investigate generalized prime hyperideals in the framework of Krasner hyperrings. To this end, new classes of hyperideals are introduced and analyzed based on multiplicatively closed properties. In particular, the concepts of s-m-hypersystems, f-hypersystems, and their associated s-prime and f-prime hyperideals are defined and examined. A subset S⊆R of a Krasner hyperring is called an s-m-hypersystem if, for every s∈S, there exists a multiplicatively closed subset S∗⊆S such that ⟨s⟩∩S∗≠∅. This concept extends the classical ideal of multiplicative compatibility to the setting of hyperrings. Furthermore, for each element a∈R, we define a hyperideal f(a) satisfying the following conditions: (i) a∈f(a), (ii) For any hyperideal K, if x∈f(a)+K, then f(x)⊆f(a)+K. Using this notion, a subset S⊆R is defined to be an f-hypersystem if there exists a multiplicatively closed subset S∗⊆S such that f(a)∩S∗≠∅ for every a∈S. We provide characterizations and original examples of these hypersystems and their corresponding prime hyperideals. The relationships and distinctions between the s-m-hypersystems and f-hypersystems are also explored. Our findings offer a refined perspective on hyperideal theory and open new pathways for the algebraic analysis of hyperstructures.