Co-first modules

This paper explores the concept of $\mathbf{\text{co-first}}\mathbf{\text{modules}}$, 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 $\mathcal{A}\mathbf{\text{-co-first}}\mathbf{\text{modules}}$ and $\mathcal{A}\mathbf{\text{-fully}}\mathbf{\text{co-first}}\mathbf{\text{modules}}$, 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.