Graded ϕ -2-Absorbing and Graded ϕ -2-Absorbing Primary Submodules

The main goal of this article is to explore the concepts of graded ϕ -2-absorbing and graded ϕ -2-absorbing primary submodules as a new generalization of the concepts of graded 2-absorbing and graded 2-absorbing primary submodules. Let ϕ : G S ( M ) → G S ( M ) ⋃ { ∅ } be a function, where G S ( M ) denotes the collection of graded R -submodules of M . A proper K ∈ G S ( M ) is said to be a graded ϕ -2-absorbing R -submodule of M if whenever x , y are homogeneous elements of R and s is a homogeneous element of M with x y s ∈ K − ϕ ( K ) , then x s ∈ K or y s ∈ K or x y ∈ ( K : R M ) , and we call K a graded ϕ -2-absorbing primary R -submodule of M if whenever x , y are homogeneous elements of R and s is a homogeneous element of M with x y s ∈ K − ϕ ( K ) , then x s or y s is in the graded radical of K or x y ∈ ( K : R M ) . Several properties of these new forms of graded submodules are investigated.