As articles on this subject indicate, a smoke ring is a combination of nitrogen, oxygen, moisture, and myoglobin. the use of high temperature combustion to produce nitrogen dioxide along with the moisture and myoglobin in the meats will produce nitrous acid which creates the smoke ring. A smoldering fire without flame will not produce as much nitrogen dioxide as a hot wood fire.
The conditions that help contribute to the development of a smoke ring are slow cooking and smoking over several hours allowing time for the nitrogen dioxide to be absorbed into, and interact with the meat pigment. As nitrogen dioxide is water soluble, maintaining a moist meat surface allows more absorption than a dry surface. To facilitate this phenomenon, marinating the product or using a water pan or burning green wood (added benefit, green wood produces more nitrogen dioxide) or soaked chips helps maintain the humidity level in the smoker. As we all know, don't open the door, this lets out the heat and mositure, causes a temperature spike and increases cooking time.