Let R be a commutative semiring with identity and M be a unitary R- semimodule. Let φ : S(M ) → S(M ) ∪ {∅} be a function, where S(M ) is the set of all subsemimodules of M . We say a proper subsemimodule N of M is φ-primary subsemimodule, if r ∈ R, x ∈ M and rx ∈ N − φ(N ) imply that r ∈ √(N :R M ) or x ∈ N . The notion of φ-primary subsemimodules is a generalization of the concept of primary, weakly primary and φ-prime subsemimodules. We study properties of φ-primary subsemimodules of a semimodule M and related results to those of ring theory.
