On the number of ring topologies on countable rings

For any countable ring $R$ and any non-discrete metrizable ring topology $\tau_0$, the lattice of all ring topologies admits:
– Continuum of non-discrete metrizable ring topologies stronger than the given topology $\tau_0$ and such that $\sup\{\tau_1,\tau_2\}$ is the discrete topology for any different topologies;
– Continuum of non-discrete metrizable ring topologies stronger than $\tau_0$ and such that any two of these topologies are comparable;
– Two to the power of continuum of ring topologies stronger than $\tau_0$, each of them being a coatom in the lattice of all ring topologies.