SOME RESULTS ON ϕ -(k,n)-CLOSED SUBMODULES

Abstract

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 $phi :S(M)rightarrow S(M)cup lbraceemptysetrbrace$ be a function. A proper submodule $N$ of $M$ is called $phi$ -semi-$n$-absorbing if $r^{n} min Nsetminus phi(N)$ where $rin R, min M$ and $nin {Bbb Z}^+$, then $r^{n} in (N:M)$ or $r^{n-1} min N$. Let $k$ and $n$ are positive integers where $k>n$. A proper submodule $N$ of $M$ is called $phi$ -$(k,n)$- closed submodule, if $ r^{k}min Nsetminus phi(N)$ where $rin R$, $min M$ and $kin {Bbb Z}^+$, then $r^{n}in (N:M)$ or $r^{n-1}min N$. In this work, firstly, we will study some general results when we use the definition $phi$ -$(k,n)$- closed submodule. Moreover, we prove main results of the $phi$ -$(k,n)$- closed submodule for various modules.