Some remarks on generalizations of classical prime submodules

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. Suppose that $phi:S(M)rightarrow S(M)cup lbraceemptysetrbrace$ be a function where $S(M)$ is the set of all submodules of $M$. A proper submodule $N$ of $M$ is called an $(n-1, n)$-$phi$-classical prime submodule, if whenever $r_{1},ldots,r_{n-1}in R$ and $min M$ with $r_{1}ldots r_{n-1}min Nsetminusphi(N)$, then $r_{1}ldots r_{i-1}r_{i+1}ldots r_{n-1}min N$, for some $iinlbrace 1,ldots, n-1rbrace$ $(ngeqslant 3)$.In this work, $(n-1, n)$-$phi$-classical prime submodules are studied and some results are established.