Preradicals and characteristic submodules: connections and operations

Abstract

For an arbitrary module \(M\in R\)-Mod the relation between the lattice \(\mathbf{L}^{ch}(_{R}M)\)  of characteristic (fully invariant) submodules of \(M\) and big lattice \(R\)-pr of preradicals of \(R\)-Mod is studied. Some isomorphic images of \(\mathbf{L}^{ch}(_{R}M)\)  in \(R\)-pr are constructed. Using the product and coproduct in \(R\)-pr four operations  in the lattice  \(\mathbf{L}^{ch}(_{R}M)\) are defined. Some properties of these operations are shown and their relations with the lattice operations in \(\mathbf{L}^{ch}(_{R}M)\)  are investigated. As application the case \(_{R} M =  _{R} R \) is mentioned, when \(\mathbf{L}^{ch}(_{R}R)\) is the lattice of two-sided ideals of ring \(R\).