on (n − 1, n)-φm-prime and (n − 1, n)-weakly prime submodules

Let m, n ≥ 2 be two positive integers, r a commutative ring with identity and m a unitary r-module. a proper submodule p of m is an (n − 1, n)-φm-prime ((n − 1, n)-weakly prime) submodule if a1, . . . , an−1 ∈ r, x ∈ m with a1 . . . an−1x ∈ p (p : m )m−1p (0 ̸= a1 . . . an 1x ∈ p ) implies a1 . . . a i−1 ai+1 . . . a n−1x ∈ p , for some i ∈ {1, . . . , n − 1} or a1 . . . an−1 ∈ (p : m ). in this paper we study these type of submodules, giving some useful results and examples concerning them.