Since you can't seem to follow rules... see the RED text at the top of the screen ("WHEN POSTING, PLEASE INCLUDE YOUR STATE BECAUSE LAWS VARY FROM STATE TO STATE."), any answer you get will be generic.
You are generally obligated to keep any property that was PROPERLY left at your house for as long as it takes you to PROPERLY make every reasonable effort to return it. That means that you GENERALLY have to send a letter (Certified) to his last known address spelling out:
1) that he needs to arrange for pickup of his property, and
2) giving a reason amount of time to contact you (7 days?), and
3) that if he fails to do so, you will dispose of the 'abandoned' property in accordance with the laws of the state of ______________ (usually by auction) and send the proceeds to him, less any allowed costs.
As to the issue of the 'ring', whose name is on the receipt, that will determine the owner. However, if as you claim, you made the payments, if it were me, I would keep the ring and let him litigate the issue, if he wants.