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 $\phi :S(M)\rightarrow S(M)\cup \lbrace\emptyset\rbrace$ be a function. A proper submodule $N$ of $M$ is called $\phi$ -semi-$n$-absorbing if $r^{n} m\in N\setminus \phi(N)$ where $r\in R, m\in M$ and $n\in {\Bbb Z}^+$, then $r^{n} \in (N:M)$ or $r^{n-۱} m\in N$. 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}m\in N\setminus \phi(N)$ where $r\in R$, $m\in M$ and $k\in {\Bbb Z}^+$, then $r^{n}\in (N:M)$ or $r^{n-۱}m\in N$. 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.

Keywords:

Ф-(k , n)-Closed submodule , Ф-Semi-n-absorbing submodule , Ф-Prime submodule