Steel wool will not help if the 'ring' is below the topcoat. Steel wool flattens the topcoat and smooths it, so its ideal to take off paint from moving and bumping the walls, accumulated grunge and hairline scratches, and leveling a finish.

If the finish is lacquer, then a ring below the surface has penetrated the topcoat and you can only get that out with a product called "No Blush", which 'opens up' the topcoat and lets the humidity / moisture causing the blush to escape, the closes up as it dries. 90% of the time that's heat - not water damage. When a hot item is placed atop the surface, it opens the finish and moisture gets absorbed from the heat. As it cools off, the topcoat closes up again. Ever put a hot plate on a wooden surface and then notice a light stick to it when you clear the dishes later on? That's a heat damaged topcoat. With No Blush, you can open the topcoat, allow it to adjust, then it will seal back up and finish with a light recoat of the lacquer and steel wool it to level. I've done it many times.

# 1 cause of heat damage is Pizza Boxes on the table top. Keep 'em off the table!