What can I do about a Ring on wood furniture?

Try rubbing plain white toothpaste or any product of an oily substance in a circular motion with a soft cloth.

I would be concerned about an abrasive substance in toothpaste. An old remedy is to use mineral oil and let it set for a couple of hours and then massage it into the wood in a circular pattern. Mineral oil doesn't turn rancid as some other oils or mayonnaise might. We all know rancid oil really smells bad.

use a little mayonnaise

Buy some, Old English. I go over ALL my wood furniture with it about once a year. It deepens the color and restores the shine and beauty.

Olive oil, or food grade mineral oil that you use on butcher blocks, or wooden salad bowls, etc.