  We show that every proper, dense ideal in a C*-algebra is contained in a
prime ideal. It follows that a subset generates a C*-algebra as a not
necessarily closed ideal if and only if it is not contained in any prime ideal.
  This allows us to transfer Lie theory results from prime rings to
C*-algebras. For example, if a C*-algebra $A$ is generated by its commutator
subspace $[A,A]$ as a ring, then $[[A,A],[A,A]] = [A,A]$. Further, given Lie
ideals $K$ and $L$ in $A$, then $[K,L]$ generates $A$ as a not necessarily
closed ideal if and only if $[K,K]$ and $[L,L]$ do, and moreover this implies
that $[K,L]=[A,A]$.
  We also discover new properties of the subspace generated by square-zero
elements and relate it to the commutator subspace of a C*-algebra.
