This happens a lot, when rings are simply embedded into the plaster while it's setting up. The problem is, plaster and metal don't get along too well.
Get your hands on some Durham's Water Putty. It's a wood putty you can mix strong or weak, depending on the job.
Take a small drill bit and drill a hole, starting with where the ring originally pulled out, and make an angle hole. Like leaning off to the side. I'd drill a hole off to the left and the right, to provide a stronger anchor. Drill out the hole the ring left behind, to make it a little bigger.
You can use the same rings or just get some strong wire and loop it.
Use a syringe to inject the water putty down into the freshly drilled holes, then pack water putty into the larger hole where you're going to insert a ring (or loop). Make sure there's some sort of anchor in your ring or loop. Like say you're using wire - twist it so that you have a ring but then also bend up the loose ends to form a hook.Wire with nothing more than a straight end will eventually let loose. Wire with hooks ends would have to cut through the dried putty. Makes it stronger.
You will have to make a fairly loose mixture of putty to inject it but then when you get to the large center hole (where the ring came out), mix it as thick as you can, then pack it in tight. The thicker, the stronger.
The drilled holes off to the side, form legs to your patch job, making that stronger too.
This is called a restoration job.