One mongo molecule to rule them all.
I will venture here into heady speculation, because I want a liquid ring.
A liquid should have a boiling point. The higher the molecular weight, the higher the barrier to turn these molecules to gas. Consider hydrocarbons. Low molecular weight hydrocarbons are volatile, like gasoline. High molecular weight hydrocarbons are nonvolatile, like asphalt. For really big polymers the heat necessary to boil is in excess of that which breaks down the intermolecular bonds - especially in the presence of oxygen.
So: we can have long polymers which do not boil. I propose the ring be made of such molecules. The ring may, in fact, be one enormous planet-girdling cross linked polymeric molecule. Boilproof. Are enormous macroscopic single molecules possible? Go look at your car's tires. How big could a car tire be?
But it must be liquid. That can be achieved by tweaking the flexibilty of the main chain and the strength of the crosslinks between loops of this asphalt like polysilane-like chain. Heat will be imparted by any nearby heat source, like a star or the planet. With no volatility, shedding heat will rely on radiation alone, and this giant polymer ring in orbit will become very warm. Warmed to a fluctuant fluidity!