Well..They Will Switch But Only In The Place THEIR In. :/ I Needs Help. Inventory To Inventory And Bar To Bar.
Okay, like Gimpi00 and spynoodle said, you have to be in the Null Chamber, the place you end up when you die, to move a ring from Bar to Bag/Bag to Bar.
However, IF you have an empty slot in your bar, you can place a ring from the bag in it without being in the Null. Bar to Bag only works in the Null.
Also, you can never wear two copies of the same ring.