Krull's principal ideal theorem

Let $A$ be a Noetherian ring and $f\in A$ . If a prime ideal $\mathfrak p$ is minimal among the prime ideals containing $(f)$ , then

\operatorname{ht}(\mathfrak p)\le 1.

Here $\operatorname{ht}(\mathfrak p)$ is the height of $\mathfrak p$ .

Geometric form

In the prime spectrum $\operatorname{Spec}(A)$ , every irreducible component of $V(f)$ has codimension at most $1$ .

Generalization

Krull's height theorem states that if an ideal $I$ can be generated by $r$ elements, then every prime ideal minimal over $I$ has height at most $r$ .

Example

For a field $k$ , the prime ideals $(x)$ and $(y)$ are minimal over $(xy)$ in $k[x,y]$ , and each has height $1$ .