I want to prove that finite morphisms of schemes are closed, but I cannot prove the affine case, namely:
Given a finite morphism of rings $\varphi :B \to A$ prove that the induced morphism of schemes $f:X \to Y$ is closed.
For this I'm trying to show that $f(V(I))=V(\varphi ^{-1}(I))$, where $V(I)$ denotes the set of all prime ideals that contain $I$.
I managed to prove the above equality only when $I$ is prime (Going-Up theorem), but I am missing something when $I$ is an arbitrary ideal of $A$. Also, from this I can reach the conclusion when $A$ is noetherian, since any closed set would be a finite union of irreducible closed sets, but I cannot figure out the general case.