Krull's principal ideal theorem

Statement

Let $R$ be a Noetherian and $x\in R$. Let $P$ be a minimal prime ideal among those containing $x$. Then, the codimension of $P$ is at most 1.

Generalizations

• Krull's height theorem: This is often also called the final version of the principal ideal theorem.
• Determinantal ideal theorem: This generalizes the principal ideal theorem to the ideal generated by the determinants of minors of a matrix