Some Remarks on ϕ-Graded Semi-n-Absorbing Submodules

Let r be a g-graded commutative ring with identity and m be a unitary g-graded r-module. let s(m) be the set of all graded submodules of m and ϕ : s(m) → s(m) ∪ {∅} be a function. a proper graded submodule n of m is called ϕ-graded semi-n-absorbing submodule if whenever r ∈ h(r) and m ∈ h(m) with rnm ∈ nϕ(n), then rn ∈ (n : m) or rn−1m ∈ n (n ≥ 2). in this work, firstly, we state with deeper results on the structure of generalizations of prime submodules as ϕ-graded prime submodules. moreover ϕ-graded semi-n-absorbing submodules are studied and some results are obtained.