Let A and B be commutative rings with 1, A a subring of B. Is there a section of the inclusion, namely a ringhomomorphism from B to A, that is the identity on A? I hope not and I think that would be plaubsible. But I have no idea how to proof and I couldn't find something about it. Maybe it is easier to verify if A and B are concrete ring, A the real valued smooth and B the real valued continous functions on a differential manifold.