I recently asked an electrical engineer if the Earth's magnetic field would have any effect on a permanent ring magnet that is circumferentially-magnetized. This was his reply:
"There will be some interaction although not externally noticeable. A fully magnetized ideal permanent magnet has a permeability of vacuum, µ0 because all of its magnetic dipoles are aligned at the magnetization direction, which is a magnetic saturation. This implies that the Earth's magnetic field passes through the ring unchanged.
All the dipoles sense the external magnetic fields. we now assume that the ring’s plane is parallel to the field, i.e. that the Earth's magnetic field is parallel to a diameter of the ring. Except for the few dipoles oriented parallel to the field most dipoles will experience a torque. This internal torque is the main interaction. The symmetry of the ring and the circular magnetization direction imply a net zero torque on the magnet as a whole but the local torques inside do apply a weak bending stress on the magnet’s material.
Actual magnets do have some non-unity relative permeability, around 1.05. That 5% response is similar to any ferromagnet thus in addition to the above localized torque there is a small rotational torque on the entire ring if the ring is not perpendicular to the field or diametrically parallel to it."
I then asked him what would be the rotational speed of this ring magnet due to the '...small rotational torque on the entire ring...' and his reply was "...I assume that the magnetization is full closed loop, i.e. the field lines are a circle inside the ring. In that case the force on the ring is extremely low and probably completely negligible and only very sensitive equipment can measure it."
If the Earth's magnetic field (~ .5 Gauss) is too weak to cause any rotation within a permanent ring magnet that is circumferentially-magnetized, then perhaps placing it within a much stronger magnetic field may rotate it?
For example, if this ring magnet were to be placed between the North pole of a neodymium magnet and the South pole of another neodymium magnet, the strong magnetic field between these two poles may be strong enough to cause the ring magnet to start to rotate. The ring magnet would be put on a non-metallic shaft and this shaft would be inserted between two non-metallic shaft bearings to enable the ring magnet to rotate within this field.