---
date: 2022-01-24 14:45
modification date: Monday 24th January 2022 14:46:19
title: "Krull's principal ideal theorem"
aliases: ["Krull's principal ideal theorem", "Hauptidealsatz"]

---

Tags: 
#todo #CA  
Refs:
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# Krull's principal ideal theorem (Hauptidealsatz)

Let $R$ be a Noetherian ring and $a$ an element of $R$ which is neither a zero divisor nor a unit. Then every minimal prime ideal $P$ containing $a$ has height 1 .

![](attachments/Pasted%20image%2020220415133222.png)

![](attachments/Pasted%20image%2020220415133239.png)