  We study a particle on a ring in presence of a dissipative Caldeira-Leggett
environment and derive its response to a DC field. We find, through a 2-loop
renormalization group analysis, that a large dissipation parameter $\eta$ flows
to a fixed point $\eta_R=\eta_c=\hbar/2\pi$. We also reexamine the mapping of
this problem to that of the Coulomb box and show that the relaxation
resistance, of recent interest, is quantized for large $\eta$. For finite
$\eta>\eta_c$ we find that a certain average of the relaxation resistance is
quantized. We propose a box experiment to measure a quantized noise.
