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CorollaryStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01
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Minimal primes are exactly the primes of height zero

Statement

Let R be a commutative ring and let p∈Spec⁡(R). Then p is minimal if and only if ht⁡(p)=0.

Facts & Assumptions

Given: A commutative ring R and a prime ideal p⊂R.

[L1]

The height of p is the supremum of the lengths of strict prime chains ending at p (Height equals local dimension).

Proof

technique · direct
1.1

If p is minimal, there is no strict prime ideal properly contained in p. Therefore every strict chain ending at p has length 0, and [L1] gives ht⁡(p)=0.

L1given
1.2

If ht⁡(p)=0, then [L1] says no strict chain of positive length ends at p. In particular there is no prime ideal properly contained in p, so p is minimal.

L1given
2.1

The two implications prove that minimal primes are exactly the primes of height zero.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

4 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