  We show that a not necessarily closed ideal in a C*-algebra is semiprime if
and only if it is idempotent, if and only if it is closed under square roots of
positive elements. Among other things, it follows that prime and semiprime
ideals in C*-algebras are automatically self-adjoint.
  To prove the above, we isolate and study a particular class of ideals, which
we call Dixmier ideals. As it turns out, there is a rich theory of powers and
roots for Dixmier ideals. We show that every ideal in a C*-algebra is squeezed
by Dixmier ideals from inside and outside tightly in a suitable sense, from
which we are able to deduce information about the ideal in the middle.
