Endo Second Submodules, Endo Quasi-Prime Second Submodules, and Related Concepts

Abstract

The notions of endo-second submodules and endo quasi-prime second submodules are presented as generalizations of second submodules and quasi-prime second submodules, respectively. A nonzero submodule B of an A-module V is called an endo-second submodule if, for every f ∈ End(V), either f(B) = B or f(B) = 0. A nonzero submodule B of V is called an endo quasi-prime second submodule if, for f, g ∈ End(V) and every completely irreducible submodule S of V, the relevant inclusion implies that either f(B) ⊆ S or g(B) ⊆ S. Furthermore, a nonzero submodule B of an A-module V is called a 2-absorbing second submodule if, whenever a, b ∈ T, L is a completely irreducible submodule of V, and Bab ⊆ L, then Ba ⊆ L, Bb ⊆ L, or a · b ∈ Annₜ(B). This notion may be regarded as the dual of a 2-absorbing submodule. These concepts are defined through the action of the endomorphism ring on modules, providing a structural perspective that connects endomorphism behavior with the internal properties of modules. The study establishes several fundamental properties and theorems related to these concepts and clarifies their relationships with other classes of second submodules.