The simple answer to your question is . . . sort of. As with most things, there is no simple answer, but I've got a lot of information that might help sort things out.
Like prime rib, rib-eye is a cut of beef from the rib section. Although they come from the same part of the animal, they are cut differently and cooked differently.
A rib-eye is a steak, a piece of meat that is cut across the muscle into a thick or thin slice, with or without the bone, which should usually be cooked quickly. You will also find the rib-eye labeled with many aliases - Delmonico, rib and cowboy steak.
By the way, the word steak can refer to other meats such as lamb or venison and does not have to mean beef. (Unless you're from Texas, like I am, where talk like that can land you in the sheriff's office.)
Prime rib or rib roast is a roast - a larger piece of meat that will serve more than one person and should be cooked whole, normally in the oven.
The cut known as rib roast comes in two styles: the standing rib roast, which includes the entire roast with three to seven rib bones; and the rolled rib roast, which is boneless and tied into a tight package.
The name standing rib roast comes from the fact that it was often roasted in a standing position with the ribs sticking up. These days, most cooks prefer to cook this roast with the ribs on the bottom, which creates a natural roasting rack.
Since you asked for my expertise, Thomas, I advise buying this cut with the bones intact, because they not only act as a rack, they also add more flavor.
When cooked that way, obviously, it's not a standing rib roast anymore. Maybe that's how it got the name prime rib, although that term can be confusing.
The word "prime" in the term "prime rib" does not indicate that the piece of meat is USDA Prime grade beef.
