Prime ringsMaybe classified by the sizes of the sets that‘insulate’ their elements from annihilation. For a cardinal m > 0, the class [Pbar]r,(m) of all rings that are right prime of ‘bound at most m’ is studied, with particular reference to its closure under constructions such as matrix rings, semigoup rings, orders and extensions. The classes [Pbar]r,(m) are special in the sense of radical theory for each m > 0. The attendant upper radicals υ[Pbar]r,(m) are right (and not left) strong; their compatibility with certain ring constructions is examined. In the lattice of radicals (where they form a strictly descending chain), their positions are described, relative to various familiar radicals.