Relentless Rohirrim

The Ring gains its abilities in order from top to bottom. Once it gains an ability, it has that ability for the rest of the game.

Each time the Ring tempts you, you must choose a creature if you control one.

If the creature you choose as your Ring-bearer was already your Ring-bearer, that still counts as choosing that creature as your Ring-bearer for the purpose of abilities that trigger "whenever you choose a creature as your Ring-bearer" or abilities that care about which creature was chosen as your Ring-bearer.