Remark

If $h$ is surjective and $a$ is its kernel, it follows from the definition of the spectral topology that $^a h$ is a homeomorphism of $X'$ onto the closed subspace $V(a)$ of $X$; for a prime ideal $p'$ of $A'$ to contain an ideal $b'$ of $A'$, it is necessary and sufficient that $\overline{h}(p')$ contain $\overline{h}(b')$; we see first that $^a h$ is injective by taking $b'$ to be prime; moreover, for every ideal $b'$ of $A'$, the image under $^a h$ of $V(b')$ is $V(\overline{h}(b'))$, whence our assertion, the ideals of the form $\overline{h}(b')$ all being ideals of $A$ which contain $a$.