I just studied Theorem 2.6.3 on Kempf's "Algebraic Varieties", which is a geometric version of the principal ideal theorem. It states:
Let g be a non-zero regular function on an irreducible variety $X$. Then each component of the closed subset $(g=0)=\{x\in X|g(x)=0\}$ has dimension $\dim(X)-1$.
My main question is: what exactly links this statement to Krull's principal ideal theorem in its classical form. I know irreducible closed sets of an affine variety correspond to prime ideals in the ring of regular functions and I know this correspondance inverts inclusions, so I can translate Kempf's definition of dimension in a purely ideal-based one, where the ideal associated to the irreducible component is a minimal prime ideal. But I don't see why any of these ideals should be principal.