just to let you guys know imo i highly doubt this is the same picopico hammer for dotnuri
if you read the icon description it says"[Quest] Complete missions given by the Rocking Hammer"
you get the ring upon accepting the quest "Unexpected Greenhorn" which requires the use of the hammer to complete. once you turn in the quest you turn in the ring as well so you lose the ring when youre done with the quest
however if you accept the quest and abandon it you will keep the ring so you can still have the hammer after completing all the quests