I can't find it at the moment and I'm on my way out, but I am almost sure it costs extra to change the slot of an item. 50% more maybe? I just can't seem to find it :(
You won't find it, that was a rule from 3.5 (a sidenote iirc) that is no longer part of Pathfinder.
AfAIK, PF-RPG has got no rules for that (except a few notes in the Conversion Guide about Items that enhance Ability Scores).
3.5 would allow you to change the slot, but it may cost more, the same (or even less if the original item wasn't in the "proper" arbitrary slot) depending of the slot. The rule was for woundrous items, rings weren't in the list.
Since protection rings cost the same as protection amulets I'd say: Use the same price. Furthermore, loosing the slot for "amulet of natural armor" isn't a good thing.