Krull's height theorem

Statement

Let $R$ be a Noetherian commutative unital ring and $x_{1},x_{2},\ldots ,x_{c}$ be elements in $R$. Let $P$ be minimal among primes containing all the $x_{i}$s. Then, the codimension of $P$ is at most $c$.

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

Proof

Starting assumptions

Replacing $R$ by $R_{P}$ if necessary, we may assume that $R$ is a local ring with unique maximal ideal $P$.

In particular, we see that the ring $R/(x_{1},x_{2},\ldots ,x_{c})$ is a local Artinian ring with unique maximal ideal $P/(x_{1},x_{2},\ldots ,x_{c})$, hence $P$ is nilpotent modulo $(x_{1},x_{2},\ldots ,x_{c})$.

Main proof

Supose $P_{1}$ is a prime contained in $P$, with no primes between. Then, it suffices to show, inductively, that $P_{1}$ is minimal over an ideal generated by $c-1$ elements.