Co-first modules

TL;DR

This paper introduces and characterizes co-first modules, a generalization of coprime modules, through preradicals, and explores their properties, lattice structures, and connections to ring theory.

Contribution

It defines new classes of co-first modules using preradicals and analyzes their properties, extending classical module theory concepts.

Findings

Characterization of coprime modules via new submodule products

Introduction of $\\mathscr{A}$-co-first and fully co-first modules

Conditions under which co-first modules are second modules

Abstract

This paper explores the concept of \textbf{co-first modules}, a generalization of coprime modules, through the lens of preradicals in module theory. Building on foundational notions such as second modules and coprime modules, we introduce new submodule products and characterize coprime modules using these products. The study extends classical definitions by defining \textbf{$\mathscr{A}$-co-first modules} and \textbf{$\mathscr{A}$-fully co-first modules}, which utilize subclasses of preradicals to broaden the scope of coprimeness. Additionally, we investigate the lattice structure of conatural classes and their closure properties, providing conditions under which co-first modules coincide with second modules. The paper also examines the implications of these concepts in the context of left MAX rings and left perfect rings, offering a comparative analysis of coprimeness and secondness. Our results try to contribute to a deeper understanding of module theory and its applications, while highlighting connections between preradicals, module classes, and ring properties.