Nonclosed ideals of bounded operators play a prominent role in the theory of singular traces as developed by Dixmier, Connes and many others, and the Calkin correspondence is a powerful tool that can be used to answer many questions about nonclosed ideals in this context. For general C*-algebras, a systematic study of nonclosed ideals was initiated by Pedersen in the late 1960s, but much less is known in this broader setting. We show that a not necessarily closed ideal in a C*-algebra is semiprime (that is, an intersection of prime ideals) if and only if it is closed under roots of positive elements. Quite unexpectedly, it follows that prime and semiprime ideals in C*-algebras are automatically self-adjoint. This can be viewed as a generalization of the well-known result that closed ideals in C*-algebras are semiprime and self-adjoint. This is joint work with Eusebio Gardella and Kan Kitamura.