A Note on Semiprime Hyperideal in Ternary Hypersemirings

This research aims to define and investigate the properties of semiprime hyperideals in ternary hypersemirings. The researchers introduce the notion of semiprime hyperideal in a ternary hypersemiring and characterize it. The study has expanded by introducing the ideas of weak n-system and strong n-system in a ternary hypersemiring, and using these; the researchers characterize semiprime hyperideals. Again, the researchers introduce the prime radical β(I) of a hyperideal I in a ternary hypersemiring and obtain the important result that for a proper hyperideal of a ternary hypersemiring R, β(I) ={r ∈ R: every weak m-system in R which contains rhas a nonempty intersection with I}. Finally, the work has been concluded by introducing the concept of fully idempotent ternary hypersemirng, and using this concept. It has been proved that if S is a commutative ternary hypersemiring with hyperidentity, then S is a regular commutative ternary hypersemiring with hyperidentity.