Krull Principal Ideal Theorem

Statement

Let $R$ be a Noetherian ring and let $a \in R$. If $\mathfrak p \subset R$ is a prime ideal minimal over the principal ideal $(a)$ (i.e. $(a)\subseteq \mathfrak p$ and there is no prime strictly between $(a)$ and $\mathfrak p$), then

where $\operatorname{ht}(\mathfrak p)$ is the height of a prime .

Equivalently: in a Noetherian ring, every minimal prime over a principal ideal has codimension $\le 1$.

Cross-links

• Height and dimension: height , Krull dimension
• Geometry via primes: prime spectrum
• Ideals and generators: ideal , generated ideal

Examples

1. Integers.
   In $R=\mathbb Z$, take $a=12$. The primes minimal over $(12)$ are $(2)$ and $(3)$.
   Each has height $1$ (the only nontrivial chain is $(0)\subset (p)$), consistent with the theorem.

2. A polynomial ring.
   Let $R=k[x,y]$ (Noetherian) and $a=x$. The ideal $(x)$ is itself prime, hence it is the unique prime minimal over $(x)$.
   Its height is $1$, so $\operatorname{ht}(x)\le 1$ holds with equality.

3. A reducible principal ideal.
   In $R=k[x,y]$, take $a=xy$. The ideal $(xy)$ has minimal primes $(x)$ and $(y)$.
   Both have height $1$, so again the bound $\le 1$ holds even when $(a)$ is not prime.