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.