Preradicals and characteristic submodules: connections and operations

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