It's more like, there is neither without the other. You can't move in space without moving in time, and vice versa.

So, my slightly-headache-inducing answer: if the universe is infinite in size, then the ring is traveling through space-time just like we are. If space is finite, then the ring just hasn't reached the 'boundary' of whatever spacetime object we're contained in (what happens when it gets there is a question I'll happily leave to philosophers and quantum physicists).

In either case, there must be space for the ring to occupy while it travels, which necessitates the existence of time outside the ring.