Renters insurance (or tenants insurance) is a great idea, and fairly inexpensive. Without this, your belongings (clothes, furniture, books) are not covered at all if, say, the building burns down or the maintenance man (or your pet) breaks something. Particularly with fire/water damage that affects all your belongings, replacement costs are astronomical (a friend just went through this with a fire, unfortunately).
Anyway, back to ring insurance -- I called my insurer (USAA -- look into this if you had a parent in the armed forces; their rates are good) to ask about whether a ring would be covered. At least with my carrier, a renters insurance policy does not cover jewelry. You get a separate Valuable Property policy. It is not a rider to the renters policy (unlike the rider I got for my computer), but a separate policy. Don''t know if all insurers work that way.