Modules with finitely many small submodules

Abstract

O. A. S. Karamzadeh et al. [On rings with a unique proper essential right ideal, Fund. Math. 183 (2004) 229–244] introduced a class of modules with unique proper essential submodule, such modules are called as ue-modules. A. Azarang and F. Shahrisvand [Rings with only finitely many essential right ideals, Comm. Algebra 47 (2019) 2843–2854] introduced the idea of fe-module and called a module with finitely many essential submodules as a fe-module. Here, we introduce and study the classes of modules dual to them. The class of modules with finitely many small submodules is defined as fs-modules and modules with a unique nonzero small submodule are defined as us-modules.