On $\aleph_0$-Noetherian modules and $\aleph_0$-coherent modules

Abstract

In this paper, we extend the concepts of Noetherian and coherent modules to $\aleph_0$-Noetherian and $\aleph_0$-coherent modules. A module is defined as $\aleph_0$-Noetherian if every submodule is countably generated, and $\aleph_0$-coherent if it is countably generated and every countably generated submodule is countably presented. We investigate the fundamental properties of these modules and explore their behavior under several constructions, such as trivial ring extensions, fiber products, and amalgamations. Additionally, we demonstrate that $\aleph_0$-Noetherian and $\aleph_0$-coherent modules are closed under localization and provide new insights into the structure of these modules in comparison to their classical counterparts.