Subidempotent radical classes

We show that a semisimple class of rings M satisfies condition (β) (i.e. I ◃ A ◃ M and A²= 0 implies A/I ε M) if and only if the corresponding radical class is hypersolvable or hypo-idempotent. Any radical class R which satisfies condition (F) (i.e. J ◃ I ◃ A and I/J ε R implies J ◃ A) must by hypo-idempotent. If the radical class is regular, the converse is also true. We also give characterizations of the semisimple classes of hypo-idempotent and subidempotent radical classes.