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 by a 2-loop renormalization group analysis that a large dissipation parameter scales into a fixed point. We also reexamine the mapping of this problem to that of the Coulomb box and show that a certain average of the relaxation resistance, of recent interest, is quantized for large dissipation. We propose an experiment to detect this quantization.