on weakly 1-absorbing prime subsemimodules: triple-zero phenomena

In this article we undertake a systematic investigation of two novel classes of subsemihypermodules over commutative semihyperrings, which we designate as 1- absorbing prime and weakly 1-absorbing prime subsemihypermodules. throughout, let s be a commutative semihyperring with identity 1 ̸= 0, and let m denote a left shypermodule. a proper subsemihypermodule n ⊊ m is called 1-absorbing prime provided that for all nonunit elements a, b ∈ s and every m ∈ m, the containment abm ∈ n forces either ab ∈ (n :s m) or m ∈ n. analogously, n is said to be weakly 1- absorbing prime whenever, for nonunit elements a, b ∈ s and m ∈ m, the nontrivial condition 0 ̸= abm ∈ n implies that ab ∈ (n :s m) or m ∈ n. our analysis develops a broad framework for understanding these concepts by establishing several intrinsic characterizations and by exploring their algebraic consequences. in particular, we demonstrate that a proper subsemihypermodule n of m is 1-absorbing prime if and only if for every pair of proper ideals i, j of s and for every subsemihypermodule k of m, the inclusion ijk ⊆ n necessarily yields ij ⊆ (n :s m) or k ⊆ n. furthermore, it is shown that if n is a proper subtractive subhypermodule of m, then n is weakly 1-absorbing prime precisely when for every pair of proper ideals i, j ⊆ s and every subhypermodule k ⊆ m, the nontrivial containment 0 ̸= ijk ⊆ n ensures that either ij ⊆ (n :s m) or k ⊆ n. these findings serve to generalize classical notions of primeness and absorbing properties from the realm of semimodules into the broader and more intricate setting of semihypermodules. they also reveal how hyperstructural algebra intertwines with prime-like axioms, shedding light on the deeper absorptivity phenomena that arise uniquely within hyperalgebraic systems.