It is a tough property law question, but the equities may be in favor of the current holder (beachrunner) of this lost treasure trove - if it can be considered abandoned (which determination depends on a couple of more factors still)...
anyhow, I'm sure if someone is the lawful owner, and can prove that to beachrunner, that a reward ought to be offerred. Since finding the owner is unlikely, there is no duty in this case to take this ring to the sheriff, as it was unlikely to be found easily by whoever lost it last.