Prime rib frequently refers to the cooked product, regardless of grade. Just because you order prime rib from a restaurant doesn't mean it's USDA prime--more likely it's choice. So after cooking it, you can call it prime rib.
At the store you'll find: Rib roast: typicaly boneless Standing rib roast: typlically with the bones attached (whether pre-trimmed and tied back on or left intact). Ribeye roast: another name for one of the above.
You can get more specific, as to the smaller and larger end, which numbered rib bones are included in the cut, etc. But just look for a rib roast, bone in or out is your choice, that is the size you want and that has the most marbeling.
Bone-in roasts don't need a rack to cook since the bones serve this function. Boned roasts benefit from being cooked on a rack.
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