On (n − 1, n)-ϕ-second submodules

Abstract

Let R be a commutative ring with identity, M be an R-module, n ≥ 2 be a positive integer and ϕ : S(M) −→ S(M) be a function where S(M) is the set of all submodules of M. In this paper we introduce and study the concept of (n − 1, n)-ϕ-second submodule. We call a non-zero submodule N of M as an (n − 1, n)-ϕ-second submodule if (a1...an−1)N ⊆ K and (a1...an−1)ϕ(N) 6⊆ K, where a1, ..., an−1 ∈ R and K is a submodule of M, imply either a1...an−1 ∈ annR(N) or (a1...ai−1ai+1...an−1)N ⊆ K for some i ∈ {1, ..., n − 1}. We give a number of results concerning this submodule class. We characterize modules with the property that for some ϕ, every non-zero submodule is (n−1, n)-ϕ-second. We show that under some assumptions strongly (n − 1)-absorbing second submodules and (n − 1, n)-ϕ-second submodules coincide. We also focus on (2, 3)-ϕ-second submodules and give some special results concerning them.