On Generalization of prime submodules

Let $R$ be a commutative ring with identity and $M$ be a unitary‎ ‎$R$-module‎. ‎Let $\phi:S(M)\rightarrow S(M)\cup\{\emptyset\}$ be a‎ ‎function‎, ‎where $S(M)$ is the set of submodules of $M$‎. ‎Suppose‎ ‎$n\geq 2$ is a positive integer‎. ‎A proper submodule $P$ of $M$ is‎ ‎called $(n-1,n)-\phi$-prime‎, ‎if whenever $a_1,\dots,a_{n-1}\in R$‎ ‎and $x\in M$ and $a_1\dots a_{n-1}x\in P\backslash\phi(P)$‎, ‎then‎ ‎there exists $i\in\{1,\dots,n-1\}$ such that $a_1\dots‎ ‎a_{i-1}a_{i+1}\dots a_{n-1}x\in P$ or $a_1\dots a_{n-1}\in(P:M)$‎. ‎In this paper we study $(n-1,n)-\phi$-prime submodules $(n\geq‎ ‎2)$‎. ‎A number of results concerning $(n-1,n)-\phi$-prime‎ ‎submodules are given‎. ‎Modules with the property that for some‎ ‎$\phi$‎, ‎every proper submodule is $(n-1,n)-\phi$-prime‎, ‎are‎ ‎characterized and we show that under some assumptions‎ ‎$(n-1,n)$-prime submodules and $(n-1,n)-\phi_m$-prime submodules‎ ‎coincide ($n,m\geq 2$)‎.