On (n ? 1, n)-?-second submodules

Abstract

Let R be a commutative ring with identity, M be an R-module, n ? 2 be a positive integer and ?: 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)-?-second submodule. We call a non-zero submodule N of M as an (n ? 1, n)-?-second submodule if (a1 …an?1)N ? K and (a1 …an?1)?(N) ? 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 ?, every non-zero submodule is (n ? 1, n)-?-second. We show that under some assumptions strongly (n ? 1)-absorbing second submodules and (n ? 1, n)-?-second submodules coincide. We also focus on (2, 3)-?-second submodules and give some special results concerning them. © Palestine Polytechnic University-PPU 2023.