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 submodules of M. A proper submodule N of M is called an (n−1,n)-ϕ-classical prime submodule, 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.