$\phi$-classical prime submodules

Abstract

In this paper, all rings are commutative with nonzero identity. Let $M$ be an $R$-module. A proper submodule $N$ of $M$ is called a classical prime submodule, if for each $m\in M$ and elements $a,b\in R$, $abm\in N$ implies that $am\in N$ or $bm\in N$. Let $\phi:S(M)\to S(M)\cup{\emptyset}$ be a function where $S(M)$ is the set of all submodules of $M$. We introduce the concept of "$\phi$-classical prime submodules". A proper submodule $N$ of $M$ is a $\phi$-classical prime submodule if whenever $a,b\in R$ and $m\in M$ with $abm\in N\backslash\phi(N)$, then $am\in N$ or $bm\in N$.