some results on ϕ -(k,n)-closed submodules

Let r be a commutative ring with identity and m be a unitary r -module. let s(m) be the set of all submodules of m and ϕ : s(m) → s(m) ∪ {∅} be a function. a proper submodule n of m is called ϕ -semi-n-absorbing if r^nm ∈ n ϕ(n) where r ∈ r, m ∈ m and n ∈ z^+, then r n ∈ (n : m) or r^n−1m ∈ n. let k and n are positive integers where k > n. a proper submodule n of m is called ϕ -(k, n)- closed submodule, if r km ∈ n ϕ(n) where r ∈ r, m ∈ m and k ∈ z^+, then r^n ∈ (n : m) or r^n−1m ∈ n. in this work, firstly, we will study some general results when we use the definition ϕ -(k, n)- closed submodule. moreover, we prove main results of the ϕ -(k, n)- closed submodule for various modules.