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A minimal prime over a principal nonzerodivisor has height one

Statement

Let $R$ be a Noetherian commutative ring, let $x \in R$ be a nonzerodivisor, and let $p$ be a prime ideal minimal over $(x)$ . Then $ht (p) = 1$ .

Facts & Assumptions

Given: A Noetherian commutative ring $R$ , a nonzerodivisor $x \in R$ , and a prime ideal $p$ minimal over $(x)$ .

[L1]

Every prime minimal over a principal ideal has height at most $1$ (Krull's principal ideal theorem).

[L2]

The principal-ideal reduction passes to a Noetherian local domain whose maximal ideal is minimal over the image of $x$ (Reduce the principal ideal theorem to a Noetherian local domain).

[L3]

A Noetherian local domain has dimension zero exactly when it is a field (A Noetherian local domain has dimension zero exactly when it is a field).

Proof

technique · direct

1.1

By [L1], $ht (p) \leq 1$ .

L1given

2.1

Apply [L2] to a minimal prime $q \subseteq p$ of $R$ . Because $x$ is a nonzerodivisor, $x \notin q$ , so $q \neq p$ . In the reduced local domain $A = (R / q)_{p / q}$ , the image of $x$ lies in the maximal ideal. If that maximal ideal had height $0$ , then [L3] would make $A$ a field, forcing $x / 1$ to be a unit, contradiction. Hence the maximal ideal of $A$ has height $1$ , so $p$ has height at least $1$ .

L2L3step 1.1given

3.1

Steps 1.1 and 2.1 give $ht (p) = 1$ .

step 1.1step 2.1∎

Depends on

Used by

Choose a parameter that misses the top-dimensional minimal components Lemma

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., §21 (standard reference, not scraped)
J. S. Milne, A Primer of Commutative Algebra, v4.03, §21 (standard reference, not scraped)
Melvin Hochster, Dimension theory and systems of parameters (standard reference, not scraped)