φ-weakly second submodules

Let R be a commutative ring with identity and M be an R-module. A non-zero submodule N of M is said to be a weakly second submodule if rsN⊆K, where r,s∈R and K is a submodule of M, implies either rN⊆K or sN⊆K. In this paper we introduce and study the concept of φ-weakly second submodules which are generalizations of weakly second submodules. Let φ:S(M)→S(M) be a function where S(M) is the set of all submodules of M. A non-zero submodule N of M is said to be a φ-weakly second submodule if, for any elements a,b of R and a submodule K of M, abN⊆K and abφ(N)⊈K imply either aN⊆K or bN⊆K. We give some properties and characterizations of φ-weakly second submodules and investigate their relationships with weakly second submodules. M is said to be a comultiplication R-module if for every submodule N of M there exists an ideal I of R such that N=(0:M I) where (0:M I)={m∈M:Im=(0)}. We determine φ-weakly second submodules of a comultiplication module. A non-zero submodule N of M is said to be a φ-second submodule if, for any element a of R and a submodule K of M, aN⊆K and aφ(N)⊈K imply either N⊆K or aN=(0). φ-weakly second submodules are also generalizations of φ-second submodules. As a special case we prove that the concept of φ-weakly second submodule coincides with φ-second submodules when M is a comultiplication R-module. Let R=R1×R2, M=M1×M2 where Ri is a ring, Mi is an Ri-module for i=1,2. We investigate the structure of φ-weakly second submodule of the Rmodule M=M1×M2 where M1 and M2 are R-modules.