baking soda or alka seltzer or lemon Koolaid... any of these used on a sponge to wipe on the ring.... then let sit 15 minutes or so then should come off...
or fill vase with water and drop in an alka seltzer let soak awhile and then should scrub off easily..
denture cleaner tabs, drop them in fill with water..let it be overnite, rinse with cold...wala!
White vinegar also works well if wiping it on doesn't help try soaking in a bowl upside down
Try a squeezed piece of lemon or lemon juice, baking soda and vinegar, just mix in a small bowl and it will fizz up. Very good and easy to have all items on hand for all kinds of cleaning jobs.
Any of the above mentioned cleaners, then wiped out with used dryer sheets.