RE: "Re-rings"
That is short for Reinforcement Rings and they are the "extra" strips of wood that encircle the ends of the shells on the shell's interior and help to amass the bearing edge profile.
When you are talking Ludwig, in particular, it is the vintage "3-ply" shells that are equipped with the re-rings. The re-rings on vintage Ludwig 3-ply shells are made from solid maple. They are added to the very thin 3-ply shells to add strength and stability so that, when put under tension, the shell can better resist the "warping" forces loaded on it by the lugs/rims. At the same time, it also allows the main body of the shell to remain thin and more "vibrant".
3-ply shells are often less-than-perfectly round because of the way they were laid up and formed -lots of "hands-on" work. Shells of this design became too labor-intensive to make and were outmoded by more modern, thicker UNreinforced shells (1976-ish?) made with more efficient machinery.
It's always easy to tell a Ludwig 3-ply shell at a glance once you've seen a few. I have several 3-ply sets and I have taken them all apart and cleaned them. In doing so, I was amazed at how lightweight they are -especially when compared to a more modern, thicker shell.