### Generalizations Of Second Submodules

#### Abstract

In this paper, we introduce and study some new generalizations of second submodules via a function ϕ on the set of all submodules of a module. Let R be a ring with non-zero identity, M be an R-module and ϕ:S(M)→S(M) be a function where S(M) is the set of all submodules of M. A non-zero submodule N of M is said to be a ϕ-second submodule if, for any element a of R and a submodule K of M, aN⊆K and aϕ(N)⊆K imply either N⊆K or a∈ann_{R}(N). Let n≥2 be an integer and ϕ_{n}:S(M)→S(M) be the function defined by ϕ_{n}(L)=(L:_{M}ann_{R}(L)ⁿ⁻¹) for every L∈S(M). Then a ϕ_{n}-second submodule of M is said to be an n-almost second submodule of M. We determine various algebraic properties of these submodules and investigate their relationships with other known submodule classes such as second, prime and semisimple submodules. We study the structure of n-almost second submodules of modules over ZPI-rings and Dedekind domains. We also give some characterizations of modules and submodules by using n-almost second submodules.

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