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.