It depends on what you mean by "ring of infinite circumference", because you're going to have to discard some conventional ideas associated with a "ring" in ord

It depends on what you mean by "ring of infinite circumference", because you're going to have to discard some conventional ideas associated with a "ring" in order to make the definition work. If you just mean "the rooms extend infinitely in both directions", then yes, the same reasoning applies; we'll just have an arbitrary "room 0", and then "room -1" on one side and "room 1" on the other. Then you tell everyone in a negative room to move to room n-1 and everyone in a positive room to move to room n+1. However, this definition does away with the idea of a ring meeting itself on the other side. We cannot say that "room -∞" is the same as "room ∞" for example, because there neither of those rooms actually exist.