Krull's principal ideal theorem

Remarks

After localizing at $P$, one may assume that $R$ is local with maximal ideal $P$, and then $R/xR$ is Artinian. If $Q \subsetneq P$ is prime, the chain of ideals $xR + Q^{(n)}$ stabilizes, where $Q^{(n)}$ is the symbolic power; with Nakayama's lemma this gives $Q^{(n)} = Q^{(n+1)}$ for large $n$. Then $Q^n R_Q = 0$ in $R_Q$, so $R_Q$ is Artinian, and $Q$ has height $0$. For example, in $k[x, y]$ the prime $(x)$ is minimal over $x$ and has height $1$. The Noetherian hypothesis is needed. In a valuation domain whose value group is $\Z \times \Z$ with the lexicographic order, the maximal ideal has height $2$ but is minimal over a single element. If $x$ is not a zero divisor and not a unit, every prime minimal over $x$ has height exactly $1$.

Going deeper

For a student

The theorem, the Hauptidealsatz, is the base case of Krull's height theorem for ideals generated by $n$ elements. Through that generalisation, it shows that every prime ideal of a Noetherian ring has finite height. It also gives the criterion that a Noetherian domain is a unique factorisation domain exactly when every height one prime is principal. It is a basic tool in dimension theory, for example in the dimension theorem for local rings.