You remove the stones before sending them in to refineries, although some will return them to you.
On a $2000 ring found at the beach the value is in the gold and how much gold there is, there is next to no resale value in the stones.
Gold content
10k = 41.7% gold
12k = 50% gold
14k = 58.3% gold,
18k = 75% gold.
22k = 91.67% gold
24k = 99.9% gold
There are gold calculators on line you can use to get gold value, multiply the weight of the ring times k value based on spot price of gold.
Here is link to one.
Gold calculator integrated with the live gold price to calculate an accurate value before buying or selling. Features include tips on selling, gram scale options, and links to credible resources.
www.goldcalc.com
Most refineries will not deal with a single ring, not worth the trouble so save and cash in multiple rings at same time, the different Karots want matter, they will melt them down together, then test the k of them melted down and pay a percentage of that. I want except less than 90% of gold value so ask what they are paying before you have it melted.