# Krull's height theorem

From Commalg

## Statement

Let be a Noetherian commutative unital ring and be elements in . Let be minimal among primes containing all the s. Then, the codimension of is at most .

There is also a converse of this statement viz converse of Krull's height theorem.

## Proof

### Starting assumptions

Replacing by if necessary, we may assume that is a local ring with unique maximal ideal .

In particular, we see that the ring is a local Artinian ring with unique maximal ideal , hence is nilpotent modulo .

### Main proof

Supose is a prime contained in , with no primes between. Then, it suffices to show, inductively, that is minimal over an ideal generated by elements.