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.