prime Subsemimodules of Semimodules over Commutative Semirings

Let R be a commutative semiring with identity and M be a unitary Rsemimodule. Let φ : S(M) → S(M) ∪ {∅} be a function, where S(M) is the set of all subsemimodules of M . A proper subsemimodule N of M is called φ-prime subsemimodule, if r ∈ R and x ∈M with rx ∈ N \φ(N) implies that r ∈ (N :R M) or x ∈ N . So if we take φ(N) = ∅ (resp., φ(N) = {0}), a φ-prime subsemimodule is prime (resp., weakly prime). In this article we study the properties of several generalizations of prime subsemimodules.