Let A be a complex Banach algebra. It is well known that the second dual A** of A can be equipped with a multiplication that extends the original multiplication on A and makes A** a Banach algebra. We show that Rad(A) (perpendicular to)(A* . A) and Rad(A**) = (A* . A)(perpendicular to) for some classes of Banach algebras A with scattered structure space. Some applications of these results are given.