Unrefinable chains when taking the infimum in the lattice of ring topologies for a nilpotent ring

A nilpotent ring $\widehat R$ and two ring topologies $\widehat\tau''$ and $ \widehat\tau*$ on $\widehat R$ are constructed such that $\widehat\tau*$ is a coatom (i.e. between the discrete topology $\tau_d$ and $\widehat\tau*$ there no exists ring topologies) and such that between $\inf\{\widehat\tau'',\widehat\tau_d\}$ and $\inf\{\widehat\tau'',\widehat\tau*\}$ there exists an infinite chain of ring topologies in the lattice of all ring topologies of the ring $\widehat R$.