CorollaryStatement: Literature-sourcedProof: Literature-sourcedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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Dimension of a quotient via chains above an ideal

Statement

Let $R$ be a commutative ring and let $I ⊴ R$ be an ideal. Assume $R / I$ is nonzero. Then $\dim (R / I) = \sup {n \geq 0 : p_{0} ⊊ \dots ⊊ p_{n} is a strict chain of prime ideals of R all containing I}$ . The supremum is allowed to be infinite.

Facts & Assumptions

Given: A commutative ring $R$ , an ideal $I ⊴ R$ , and a nonzero quotient ring $R / I$ .

[L1]

Krull dimension is the supremum of lengths of strict chains of prime ideals (Krull dimension of a nonzero ring).

[L2]

Prime ideals of $R / I$ correspond exactly to prime ideals of $R$ containing $I$ , with strict inclusions preserved (Prime ideals of a quotient ring are exactly the prime ideals containing the ideal).

Proof

technique · direct

1.1

By [L2], strict chains of prime ideals in $R / I$ are in bijection with strict chains of prime ideals of $R$ whose every term contains $I$ . Corresponding chains have the same length.

L2given

2.1

Since $R / I$ is nonzero, [L1] applies to it. Thus $\dim (R / I)$ is the supremum of the lengths of the strict prime chains in $R / I$ , and step 1.1 identifies that supremum with the one displayed in the statement.

L1step 1.1

3.1

Therefore dimension of the quotient is computed by prime chains of $R$ lying above $I$ .

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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

J. S. Milne, A Primer of Commutative Algebra, v4.03, Definition 3.14 (standard reference, not scraped)
The Stacks Project, Section 10.60: Dimension of rings (standard reference, not scraped)