Krull's Principal Ideal Theorem

Krull's principal ideal theorem is one of the most fundamental results in commutative algebra, bounding the height of a prime ideal in terms of the number of generators of an ideal it contains.

The Main Theorem

Theorem6.6Krull's Principal Ideal Theorem (Hauptidealsatz)

Let $R$ be a Noetherian ring and $x \in R$ a non-unit. If $\mathfrak{p}$ is a minimal prime over the principal ideal $(x)$ , then $\operatorname{ht}(\mathfrak{p}) \leq 1$ . Equivalently, every prime ideal minimal over a principal ideal in a Noetherian ring has height at most $1$ .

This result has a powerful generalization to ideals generated by multiple elements:

Theorem6.7Krull's Height Theorem (Generalized Principal Ideal Theorem)

Let $R$ be a Noetherian ring and $I = (x_1, \ldots, x_r)$ an ideal generated by $r$ elements. If $\mathfrak{p}$ is a prime ideal minimal over $I$ , then $\operatorname{ht}(\mathfrak{p}) \leq r$ .

The proof of the generalized version proceeds by induction on $r$ , using the principal ideal theorem as the base case.

Converse

Theorem6.8Converse of Krull's Theorem

Let $R$ be a Noetherian ring and $\mathfrak{p}$ a prime ideal of height $r$ . Then there exist elements $x_1, \ldots, x_r \in \mathfrak{p}$ such that $\mathfrak{p}$ is a minimal prime over $(x_1, \ldots, x_r)$ .

Together, the height theorem and its converse show that in a Noetherian ring, the height of any prime ideal $\mathfrak{p}$ equals the minimum number of generators of an ideal whose radical contains $\mathfrak{p}$ as a minimal prime.

ExampleHeight in quotient rings

Consider $R = k[x, y, z]/(xy - z^2)$ . The prime $\mathfrak{p} = (\bar{x}, \bar{z})$ has height $1$ (not $2$ ), since $\bar{x}$ and $\bar{z}$ are algebraically dependent in $R$ via $\bar{x}\bar{y} = \bar{z}^2$ . The principal ideal $(\bar{x})$ has $\mathfrak{p}$ as a minimal prime, consistent with $\operatorname{ht}(\mathfrak{p}) = 1$ .

RemarkGeometric interpretation

Geometrically, Krull's theorem says that a hypersurface (defined by one equation) in an $n$ -dimensional variety has pure codimension $\leq 1$ at every point. More generally, the intersection of $r$ hypersurfaces has codimension $\leq r$ at every irreducible component. This is the algebraic foundation of the dimension theory of algebraic varieties.