Current carrying conductor axially through a copper ring
Very little effect.

If the ring had a relative permeability much different than 1, then the ring of copper would increase or decrease the inductance seen by the conductor at that section. But, the relative permeability of copper is very nearly 1.

If you have AC currents of a very high frequency in the conductor, the current will couple into the ring capacitively and migrate from the center conductor to the ring by skin effect. But, "ring" implies that this ring isn't very long (otherwise, I'd call it a "tube"), so the total capacitance between the ring and the conductor is small, so the total energy coupled will be very small, and only then to the extent that you have very high frequency currents in the conductor.

Consider that many buildings are constructed with mains wiring inside metal conduit. The only noticeable effect here is added physical robustness, and this is a "ring" that extends the entire length of the cable.