When Frodo is stabbed by the Witch King of Angmar's Morgul blade at Weathertop, Aragorn explains that Frodo is slowly becoming a wraith himself, the longer he goes untreated. If Frodo had become a wraith, would he have become one of the Nazgûl -- a ring wraith -- or would he have just been in a state of existence where he was neither living nor dead, but without a specific purpose (For example, hunting down the One Ring.), to just wander aimlessly?
If his Morgul blade wound had not been healed, what kind of wraith would Frodo have become?