Krull's principal ideal theorem
Imposing one equation should drop the dimension of a space by at most one — slicing a solid with a single surface leaves you with something at most one dimension thinner, not two. Krull's principal ideal theorem is the rigorous algebraic incarnation of this geometric intuition: cutting by a single function cannot carve out a subvariety of codimension greater than one.
The theorem (the Hauptidealsatz) states: in a Noetherian ring R, if p is a prime ideal minimal among primes containing a principal ideal (a), then the height of p is at most 1. Its generalized form: if p is minimal over an ideal generated by r elements, then height(p) <= r. There is a partial converse — in a Noetherian ring any prime of height r is minimal over some ideal generated by r elements — so height is exactly the minimal number of equations needed to cut a prime out.
This single bound is the cornerstone of dimension theory: it forces dim R/(a) >= dim R - 1 for a nonzerodivisor a, gives the equality dim R[x] = dim R + 1 for Noetherian R, and underlies the fact that finitely generated algebras over a field are catenary and have well-behaved codimension. The Noetherian hypothesis is genuinely needed; without it heights can jump unexpectedly.
In k[x, y, z], the principal ideal (xy - z) is prime and the theorem guarantees it has height 1: cutting 3-space by the single equation xy = z yields a surface, codimension 1, not a curve.
One equation cuts codimension one.
The generalized version is sometimes called Krull's height theorem. It immediately implies that a Noetherian ring has finite-height primes, even if its total Krull dimension might be infinite.