Lattice of all topologies of countable module over countable rings

For any countable ring $R$ with discrete topology $\tau_0$ and any countable $R$-module $M$ the lattice of all $(R,\tau_0)$-module topologies contains:
– A sublattice which is isomorphic to the lattice of all real numbers with the usual order;
– Two to the power of continuum $(R,\tau_0)$-module topologies each of which is a coatom.