Motivated by some results and concepts from radical theory for abelian groups, we investigate the radical classesRof associative rings for which there is an infinite cardinal number α such that for every ring A and everyaεR(A)we haveaεR(B)for (i) some idealBof A with |B| < α, (ii) some accessible subringBof A with |B| < α Only the class {0} satisfies (i), while the classes other than {0} which satisfy (ii) are the classesDpof zerorings on divisible P-groups for setsPof primes. We also consider an analogous question for strict radical classes and subrings. Here the answer is much more complicated, and in fact undecidable in ZFC.