In summary, the needed ring is called the tarnished ring
, and is an random spawning item on the earth map of the game
I would recommend the adventure mode for this, as you can quickly reroll the map. Because this ring isn't a guaranteed item on every roll for the map, you have to reset the level until you get lucky and find it.
As far as I've seen, the ring can spawn anywhere on the map, so make sure to look around every corner.
If you found the ring, simply talk to Reggie and give him the ring. You will get the Scavanger perk and the achievement should pop too.