A bijection between the coatoms of the big lattice L -pr and the maximal fully invariant sublattices of L

Based on the concept of lattice preradical introduced in [7], we investigate the lattice-preradical theoretic counterparts of coprime and coirreducible lattice preradicals within the category σ[L] of all subgenerated lattices by L. We analyze the structural properties of coprime lattice preradicals, characterize them through internal operations, and explore their relation with the fully invariant sublattices of L. We also introduce the notion of coprime lattices and show how it naturally corresponds to the coprimality of their associated lattice preradicals. This connection yields significant results for complete modular lattices, including a bijective correspondence between the maximal fully invariant sublattices of a lattice L and the coatoms of L-pr, provided that L generates σ[L]. Applying these results to rings and modules, we obtain the important result that if R-pr is locally coatomic lattice, then R-pr is a set. Furthermore, we show that if M is a left R-module that generates R, then there is a bijective correspondence between the maximal two-sided ideals of R and the maximal fully invariant submodules of M. Finally, we provide the definition of a join-coprime lattice preradical and describe the relationships between this concept and the concepts of coprime and coirreducible lattice preradicals.