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 φ : s(m ) → s (m ) ∪ {∅} be a function where s(m ) is the set of all sub- modules of m . a proper submodule n of m is called an (n − 1, n)-φ-classical prime sub- module, if whenever r1, . . . , rn−1 ∈ r and m ∈ m with r1 . . . rn−1m ∈ n φ(n ), then r1 . . . ri−1ri+1 . . . rn−1m ∈ n , for some i ∈ {1, . . . , n − 1} (n ) 3). in this work, (n − 1, n)-φ-classical prime submodules are studied and some results are established.