Co-first modules
Hugo Alberto Rincón-Mejía, Luis Fernando García-Mora
TL;DR
The paper generalizes the notion of coprime $R$-modules by using preradicals to define co-first modules and their variants. It introduces a comultiplication construction $(A:B)$ and the totalizer, enabling coprimeness to be characterized via preradical-related products, and defines $\mathscr{A}$-co-first and fully $\mathscr{A}$-co-first modules relative to a subsystem $\mathscr{A}\subseteq\textbf{$R$-pr}$. It then analyzes the lattice of conatural classes $R\textit{-conat}$, CN conditions, and closes under various ring-theoretic properties, deriving connections to left MAX and left perfect rings, as well as relationships to $\mathscr{A}$-second modules. The work culminates in a detailed comparison between $\mathscr{A}$-second and $\mathscr{A}$-co-first notions, including conditions under which they coincide, thereby extending classical coprimeness and secondness in module theory. Overall, the paper broadens the framework for primality-like properties in modules through preradical machinery and explores structural and ring-theoretic implications.
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.